Wr  TO 

New 


Thomson 


WAWM08WM»WW»>yi  '■ 


««M»$«)r.W)Mnf«HIMMM(MHM«^^ 


i>?-'aw«asswi(w>>'mii>T>t>\iiwiw>wfflnry!t^^ 


OF  THE 

university 
califor^^ 


THOMSON'S     NEW    SERIES    OF    MATHEMATICS. 


KEY 


TO    NEW 


PRACTICAL  imUU 

&n  TEACH fi;^. 

JAMES    B.    THOMSON,    LL.  D., 


AUTHOR   OF   SERIES   OF  MATHEMATICS. 


NEW   YORK: 

Clark  &  Maynard,  Publishers, 

5  Barclay  Street. 
1881. 


NOTE. 


ALL  agree  that  the  best  kind  of  help  for  pupils  in 
-  Arithmetic  and  Algebra,  is  Self-hel]p ;  that  it  is 
better  for  the  learner  not  to  know  the  answer  to  a  problem, 
until  he  has  tried  his  own  ability  to  solve  it.  In  a  word, 
that  it  is  better  for  him  to  solve  a  single  example  independ- 
ently, than  a  score  by  the  help  of  a  teacher  or  a  Key. 

And  yet  it  must  be  admitted  that  a  majority  of  teachers 
desire  a  Key.  This  demand  comes  not  only  from  young 
and  inexperienced  teachers,  but  from  those  whose  char- 
acter and  scholarship  are  above  suspicion.  They  desire 
it,  not  because  of  their  inability  to  solve  the  problems, 
nor  because  they  shrink  from  labor.  Their  object  is  to 
save  time,  which  they  may  devote  to  other  branches  of 
study. 

A  well  constructed  Key  will  often  disclose  in  a  single 
minute  the  error  in  a  pupil's  work,  which  might  consume 
half  an  hour  of  the  teacher's  time,  if  he  were  obliged  to 
wade  through  a  long  operation. 

The  plan  of  the  work  before  us  is  to  indicate  in  full  the 
operations  to  be  performed,  and  give  the  results ;  omitting 
the  minor  details.  It  also  contains  many  valuable  sugges- 
tions as  to  the  different  methods  by  which  certain  problems 
may  be  solved.     It  is  hoped  teachers  will  find  it  adapted  to 

theh-  wants.  UdociMOT  IIBBJ  • 


Copyright,  1877,  by  James  B.  Thomson. 


Electrotyped  by  Smith  &  McDougal,  82  Beekman  Street. 


A      '    O 


KEY. 


EXERCISES     IN     NOTATION. 

Images  12,  18, 

g.     4C  -\-  d  -\-  7)1  —  5^:  =  ab. 

3.  5C6/  +  ^  =  xy. 

4.  ^^  +  4?^i  =  c  +  oa  —  ^ax. 

5.  «  —  h  ^  xy  ^  6mn. 

6.  a:  —  1/  +  4«  +  ^  —  ?»  =  cd  +  15m. 

2.  The  quotient  of  twice  the  product  of  a  and  h  divided 
by  X,  plus  a  minus  b,  equals  the  quotient  of  a  plus  b  divided 
by  c,  plus  the  product  of  a,  x,  and  y  minus  four  times  the 
product  of  c  and  d. 

3.  The  quotient  of  three  times  b  plus  c  divided  by  8,  plus 
3  times  X,  equals  3  times  the  product  of  c  and  d  divided  by 
a,  plus  the  product  of  x,  y,  and  z  minus  the  quotient  of  c 
divided  by  d. 

4.  The  quotient  of  3  times  a  divided  by  5,  minus  the 
product  of  a  and  x  plus  the  product  of  b  and  c,  equals  the 
quotient  of  4  times  a  minus  b  divided  by  x,  plus  the  quotient 
of  c  times  d  divided  by  4,  minus  3  times  x. 

5.  The  product  of  a,  b,  and  c  minus  x,  divided  by  3  times 
d,  increased  or  diminished  by  3  times  x  plus  5  times  ?/,  equals 
the  product  of  c,  d,  and  7i  plus  x  divided  by  twice  a,  minus 
the  product  of  x  and  y. 

M57'?055 


4  ALGEBRAIC     OPERATlOIi^S. 

6.  The  quotient  of  4  times  a  into  x  into  y  divided  by 

5  times  a,  plus  the  quotient  of  a  minus  h  divided  by  x, 
equals  the  quotient  of  x  plus  y  divided  by  a^  minus  the 

quotient  of  twice  a  plus  d  divided  by  3  times  c. 


ALGEBRAIC     OPERATIONS. 


4- 


6. 


Page  15. 

Let 

X       price  of  the  apple, 

Then  will 

2iX          "      "     '•   orange. 

And 

4.f       8  cents. 

Dividing  by 

4, 

X        2  cents,  apple;    )    , 
$x       6  cents,  orange.  ) 

And 

Let 

X       value  of  the  hay. 

Then  will 

4X           "      "     "   cow. 

And 

Sx       I40. 

Dividing  by 
And 

5. 

4X        i$^2,  cow.  ) 

Let 

X       one  of  the  numbers, 

Then  will 

^x       other  number. 

And 

4X  —  s^' 

Dividing  by 
And 

4, 

^     }  Ans. 
SX  —  27.  ) 

Let 

X       C's  number  of  peaches^ 

Then 

2X  —  B's        "         " 

And 

4X  —  A's       "        "        " 

Adding, 

yx       28. 

Dividing  by 

7, 

X         4,  C's  number;  ) 

And 

2x  —    8,  B's       "         >  Alls, 

li 

4X  —  16,  A's       "        ) 

Let 

X       son's  age. 

Then  will 

^x       father's  age. 

And 

4X       48. 

Dividing  by 

4, 

X        1 2  years,  son's  age ;     \ 
^x       36     ''      father's  age.  ) 

And 

Ans, 


ALGEBRAIC     OPF.  R  ATI  OJSTS. 


Page  16. 

8.     Let  X  =  B's  share  of  gain, 

Then  will  4X  —  A's     "        '' 

And  S^  ^^  '$100. 

Dividing  by  5,  x  ^  ^20,  B's  share  of  gain; 

And  4a;  =  ^3o,  A's     " 


a 


J. 


inSo 


Let  a;  =  ist  nnmber. 

Then  will  2X  =  2d         '* 

And  32*  =  3d        " 

Adding,  6x  =  90. 
Dividing  by  6,     x  =:  15,  ist;  ) 

And  2X  =  30,  2d;   >  Ans. 

3^'  =  45?  3^-    ^ 


ee 


1  o.     Let  X  =  value  of  the  calf, 

Then  will  Sx  —      "       "     "    cow. 

And  ()X  =  ^6;^. 
Dividing  by  9,     x  =    ^7,  price  of  calf;  ) 

And  Sx  =z  ^^6,     "      "  cow.  ) 

1 1 .  Let  X  =  value  of  bridle. 
Then  will        2X  =     "      "  saddle. 
And               21a;  =     "      '^  horse. 
Adding         24X  =:  $126. 
Divid.  by  24,    x  =      -S5.25,  worth  of  bridle; 
And                2x  =    $10.50,      '•'       "  saddle;  ]■  Ans. 

«  21X  =z  $110.25,      "       "  horse. 

12.  Let  X  =  daughters  portion, 
Then           2X  =  son's  " 
And             gx  =  wife's  " 

"  I2rc  =  836000. 

Therefore,    x  =    $3000,  daughter's  portion ; 
And  2X  =2    $6000,  son's  "         ^  A71S. 

"  gx  =  $27000,  wife's  " 


FOliCE     OF     THE     SIGKS. 


^3- 


Let 

X 

ist  number, 

Then 

3X 

2d 

a 

And 

4^  +  5 

3d 

a 

(( 

Sx-\-  5 

1877. 

Subt. 

5  from 

each,    Sx 

1872. 

Therefore, 

X 

234, 

1  st  number ; 

And 

SX 

702, 

2d 

a 

(( 

4^  +  5 

941. 

3<i 

a 

Ans. 


POWERS     AND     ROOTS. 


1-12.  Oral. 

13.  a^  -\-  IP. 

14.  (a  -\-  hy. 

15.  a  -\-  I  —  c^. 


Page  17* 

16.  \^a  -\-  "s/x. 

17.  "s/x  —  y^ 

18.  \^a  +  l\ 


ALGEBRAIC     EXPRESSIONS. 
Page  19, 


6. 

7. 
8. 


X                        6 
{a  -\-l)cd =  5  X  20 =  98^ 

c 

{x  —  a)  -\-  ax  -\ —  =  4  +  12  +  2  r=  18. 

(J/ 

x  -T-  2  -\-  (d  —  c)  -\-  dc  —  X  := 

dx  -{-  {c  —  a)  (a  —  h)  +  a;  =  30  —  2  4-  6  =  34. 

d  +  x{c  —  a)  +  «  —  a;+c  =  5  +  i24-2  — 6  +  4 


3  +  1  -f  12  —  6  = 


10. 


=  17. 


Page  21. 

I. 

60. 

2. 

40. 

3. 

«c  +  2>h. 

4. 

5^  —  2^. 

5. 

35- 

6. 

24. 

FORCE     OF    THE     SIGNS. 

8.  dh  —  7CX  4-  $a. 

9.  bxy  -\-  cxy. 

Page  22. 

iSxy 


7.     3a:  +  27/  +  fl;^». 


10. 


II. 


h  —  a 
xy 


4-  a. 
+  22. 


ADDITION. 


12.      T,x  +  xy  -\-  2ZX  sy  —  ^x  -h  xy  +  6yz,     Ans. 


Z'    i-^-y)  X 


a  —  b   ax  —  ay  —  hx  -\-  by 


Ans. 


d       ~  d 

a  =  $,     b  =  4,     c=  2,    X  =  6,    y  =  S,     and     z  =  lo. 

ax  1 8 

14.     «  H \-  yz  =  s  -\ h  80  =:  92,     Ans. 

c  2 

i^. -^  abtf 4- 2z= -  + q6-\- 20  =  120,    Ans. 

X  —  0  2 


ADDITION. 


Case  If  Page  24. '    8.   45  «5^ 


3.   2iab. 
5-   i5«^- 


9.  —  ^gabx^y^. 

10.  2g¥d7n^. 

12.  Jc  =  4. 

13-  ^«/  =  5- 


Case  II,  Page  25, 

16.  Sx. 

17.  ^Z'C. 

18.  —  12b. 

19.  —  12?/. 

20.  —  2  772, 


6.  —  2T,bcd. 

7.  —  i6x^y^. 

21.  6ab  +  i4«^  +  i5«^  +  ^^cib  —  S'^ab', 

—  'jab  —  i2ab  =  —  igab', 
Slab  —  igab  —  32«A 
Since     32^/^  =  32,     .-.     ab  =  i,    Ans. 

22.  bed  —  sbcd  +  4bcd  +  4bcd  —  sbcd  =  75. 
Uniting,  gbcd  —  8&c^  =  bed  =  75,     ^4fis. 


1.  24rz  +  2b  —  id. 

2.  1677m  —  xy  ■\-  be. 
3-       3^c 

,  —  ibc  -\-  xy  —  mn 
iibe 

EXAM 
Pagi 

PLES. 

3  26. 

4.       5«^>  - 

ab 

Zah 

—  4a'b 

ab 

4-  2Z 

gbc 

4ab  — 

■  ymn  -\-  2z,  Ans. 

i6bc  -\-  xy  —  mn,    Ans. 


8 


5-     2>^y 

—  xy  -\-  ah 

—  i^y        +  ^ 

Zxy 

—  xy 
izxy 

i^xy  -\-  ab  +  h,    Ans. 

7.  21  {a  +  b). 

8.  19^(2;  —  y). 


ADDITIOK. 

9.  jaV^y- 

10.   6\/«. 

II.   \o^/x — y. 

f»ff</e  ^7. 

dns. 

13.  a{^  —dh  -Y  3d  —  sm). 

14.  (^^  +  3  — 2c— 5^)?/. 

15.  7)1  (g  -\-  ab  —  7c  +  3d). 

16.  x{i3a—3b-}-c—3d-\-?N) 

17.  (a  +  5  —  c)2:«/. 

PROBLEMS. 


rage  28, 


X  =  cost  of  ball, 


2X  —  2  = 


i( 


a 


kite. 


^ws. 


2.  Let 
Then 

And  3X  —  2  =  46  cents. 

Adding  2  to  each  side,        3X  —  48      " 

X  =z  16     "    ball ; 

And  2x  —  2  r=  30     "    kite,  , 

Note. — As  the  learner  is  not  supposed  to  be  acquainted  with 
transposition,  he  should  in  the  operation  set  down  the  number 
required  to  be  added  to  each  side  of  the  equation,  as  seen  in  the 
solution  of  the  first  example. 

3.  Let  X  =  nnmber  of  peaches, 
Then                              2^  —  3  =       "        "  pears. 
And                               3a;  —  3  =  75. 

Adding  3  to  each  side,        3X  =  78. 

X  =:  26 peaches;  )    . 
And  2X  —  3  =  49  pears,      f 

4.  Let  X  =  the  less  nnmber; 
Then                     ^x  —  5  =  the  greater  nnmber. 
And                        6x  —  5  =  85. 

Adding  5  to  each  side,  6x  =  90. 

:r  3=  15,  the  less  No;      )  ^^^^^^ 
And  5.T  —  5  =  70,  the  greater  No.  f 


AUUITIOK.  9 

5.     Let  X  =  number  of  l)oys, 

Then  2X  —  5  :=       "        "  girls. 

And  3.T  —  5  =:  40. 

Adding  5  to  each  side,      30;  =  45. 

a;  =  15  boys  ;  )    . 
And  2a;  —  5  :=  25  gms,   ) 


6. 

44^  +  65:^  —  24         85 
44.?;  +  65  a;       109 

Uniting, 

109:?;       109 

•  •             X  —         I, 

A71S, 

7. 

7^  +  22;  —  3  —    60 

Uniting, 

9^—    63 

X  —      7, 

Ans. 

8.  4?/  +  27/  4-  5^  —  7  r=  70 

Uniting,  11?/ =  77 

y  =    y^     Ans. 

9.     Let  X  =z  B's  votes. 

Then  42^  —  20  z=  A's     *' 

And  5:^—20  =  450. 

Adding  20  to  each  side,     52^  ==  470. 

•*•        ^=    94,  B;)    ^^^ 
And  4a:  —  20  r=  356,  A,  )  ^ 

to.     Let  X  =  the  less  number. 

Then  ^x  —  3  =  the  greater  number- 

And  5a;  — 3  =z  177. 

Adding  3  to  each  side,      5:^  =  180. 

a;  =    36;  ) 
And  Ax-2>=  141,  ! 

II.  4^-}-  3^  +   2?/—   12    =   60 

Uniting,  gy  z=  72 

^  =    8,     ^7^s. 


10  ADDITION. 

12.  Let                                        a;  =  price  of  top, 
Then                           3ic  —  4  =  "       "  ball, 
xind                             4.1^—4  =  32  cents. 
Adding  4  to  eacii  side,      4X  =  36     " 

X  z=z    g  ce?its;  )     , 
And  3^  —  4  =  23     "        ) 

13.  Let  X  =  price  of  bridle, 
Then                          4^  —  5  =     "      "  saddle. 

And  s^  —  5  =  I40. 

Adding  5  to  each  side,      52:  =  $45. 

X  z=    $9,  bridle ;  ]     . 
And  4^' —  5  =  %ij  saddle,  ) 

14.  Let  X  =  sum  spent  in  a.m., 

Then  S'^  —  4  =     "        "       "  p-  m. 

4X  —  4  =  100  cents. 

Adding  4  to  each  side,      4X  =z  104     " 

X  =     26c.,  A.M. ;  )    , 
And  ^x  —  4  =z     74c.,  p.  M.,   ) 

16.  2x  4-  5'^'  +  3^  —  10  =  130 
Uniting,  loip  z=  140 

a;  =     14,     Ans, 
Proof.  28  4-  70  +  42  —  10  =  130 

17.  4X  -{-  ^x  -{-  'jx  —  12  =    86 
Uniting,  14^^==    98 

.*.        X  —      7?     jfifiSt 
Proof.  28  +  21  +  49  —  12  m    S6 

18.  loa;  —  4a;  +  93*  —  25  =  155;  .*.    :c  =  12,  ^;i6\ 
Proof.        120  —  48  +  108  —  25  r=  155 

19.  152;  —  7.C  —  2x  —  60  =z  300  ;  .'.     X  =  60,  A71S. 
Proof.        900  —  420  —  120  —  60  =  300 

20.  18.T  —  4ic  +  a;  —  75  =z  225  ;  .*.     x  =  20,  ^?i5. 
Proof.        360  —  80  +  20  —  75  =  225 


SUBTRACTION. 


11 


SUBTRACTION. 


Page  31. 


3.  14XUZ. 

4.  —  62ah. 

5.  ic^ab. 

6.  2']xy. 


7.  ^yic. 

8.  T,']ax^. 

9.  51^^^^. 

10.  —  44x'2?/^ 


11.  2)^(v^h. 

12.  o. 

13.  _  77///2.6-. 

15.  A  debt  is  propei-ly  regarded  us  a  negative  or  minus 
quantity.  Hence  we  have  to  take  —  $50  from  8100. 
$100  —  (—  $50)  =  $100  +  ^550  =  't^i5o,  Ans. 


16. 


15 

c 

10 


17. 


25°, 

$275 

_|i45 

Alls. 


20, 


^j 


Ans. 


18.  ^xy  —  6a. 

19.  13^^  +  i6a7n. 

20.  i?>x^  +  y'^  -\-  6a. 

21.  iyii-\-(l — X — 5?;z  +  3?i. 

22.  <)cd — ah — 2^4-3^  +  47/. 

23.  i8wz  —  23. 

24.  i2a;~  —  13a;. 


25.  \6ah  +  13c  +  d. 

26.  a  —  b  -j-  c. 

Proof. — Tlie  difference 
plus  the  subtrahend  equals 
the  minuend. 

rt  —  ^  +  c  =  difference. 
b — c  =  subtrahend. 

a  =  minuend. 

27.  6  (a  +  b). 

28.  g{a  —  b  -\-  x). 

29.  5  (a  +  b). 

30.  —  7  (.^'2  _  7y). 


31- 


'1000 


Then 


=  the  gain. 
Si 000  —  $500  —  $100  =  $400,  B's  share. 

$500  +  $100  =  $600,  A's     "      Ans. 


32.  If    distances    east  be   regarded   as    +,    then    those 
reckoned  west  must  be  considered  as  — . 


East  longitude, 

West 


23^ 
.37! 
60°,    Ans. 


12 


MULTIPLICATION 


Page  33. 

34.  (2b  —  c  -\-  d)  x\ 

35.  {ah  —  c  —  d  -\-  x)  y. 

36.  d^iy  —  h  -\-  c). 

37.  (cil)  —  ^c  —  d  —  m)  X. 


Ty'^.  {^  —  alj  ^  c  —  d)  xy, 
39.   ^ac  +  bmc 
2,ac  —  dc 

2ac  -\-  hmc  +  dc 

(2a  +  Im  -\-  d)  c,  AnSo 


Page  34. 

4.  h  —  (c  —  d  -{-  m)  z^h  —  c  -\-  d  —  m,    Ans. 

5.  5^;  +  ^  — «^  +  4d 

6.  2a  —  [d  -\-  c  —  {x  -{-  y)  —  d]  =^ 
2a  —  {h  -\-  c  —  X  —  y  —  d)  ^ 
2a  —  1)  —  c-\-x-{-y-\-d,     Ans. 

7.  a  —  b-^-c  —  a-\-c  +  c  —  a-\-'b. 


MULTIPLICATION. 


Case  If  Page  36. 

l^fr</f^  38. 

i*6«f/e  5,9. 

5.  42 abc. 

22. 

isxy. 

32. 

4X^y'^. 

6.  ;^^abcxy. 

^2>- 

2  4a^b^. 

33- 

2ia%^. 

7.   Sdmxy. 

24. 

a^hf. 

34. 

4oc^x"y. 

8.   6T,bcdxyz. 

25. 

d^b'"^''. 

35- 

2MW. 

9.  ^6abxy. 

26. 

Gx^z. 

36. 

—  Gx^. 

10.   42acdx. 

27. 

I  W¥c\ 

37. 

21  dWc\ 

II.   ^4bcdm. 

38. 

2%a^d^. 

12.   G^ad/xyz. 

28. 

3«  —  9 ; 

r^^       27 ; 

7—9       18,  yi;?.s. 

39- 

xhfz^. 

2: 

Page  37. 

Case  Ilf  Page  39. 

13.  Given. 

29. 

4X'       16; 

3- 

Gacx^  +  ^c^d. 

14.    —  4^abxy. 

^       4^      256; 

4. 

i^aWx—Gacdx 

15.   42  a  bed. 

256  —  16 

+  3fl!A 

16.    i^2abcxy. 

240,  Ans. 

5- 

-  ^d'bd  +  6r/^2^ 

17.    —  4i4abcxy. 

—  2bdm^. 

18.   ()4^bcdxy. 

30- 

6x^y. 

6. 

—  15^^^^ 

31- 

—  iSa'^b^c. 

+ 

20«2J2^'4-  lOfl'^cS. 

MULTIPLICATION. 


13 


Case  III,  Page  40. 


I. 


6. 


8. 


2a  +  b 

z^  +  y 


6ax  +  Tfhx  -\-  2ay  -\-  by,     Ans. 

3^  +  41/ 
a  —  b 

2,ax  +  ^ay  —  ^bx  —  ^by,     Ans. 

4b  —  c 
T,d —  a 

i2bd  —  T,ccl  —  ^ah  +  ac,     A^is. 

6xy  —  2  a 

b  -\-  c 

6bxy  —  2ab  +  6c.r?/  —  2C'c,     Ans. 
3«  +  4Z>  —  c 

^  —  y 

2,ax  4-  ^bx  —  ex  —  T,ay  —  ^by  -\-  cy,     Ans, 

5^  +  3^  4-  ^ 
a  -\-  b 

sax  +  2>'^y  +  az  +  5^.t  +  3%  +  bz,     Ans, 

']cdx  —  3rtJ 
2m  —  ^n 

i^cdmx  —  6abm  —  2\cdnx  -\-  gabn,     Ans. 

m 

Sabc  4-  4m 

3^  —  Ay 

24abcx  -\-  i2mx  —  ^2abcy  —  i6my,     Ans. 


9,  10.    Given. 

1 1 .  ;^abc"xyz''K 

12.  iiabchI"'+\ 

13.  a^X'-^". 


14.  ex  (a  +  by. 

15.  5c  {a  — by. 

16.  abc  {x  4- y)"'+''. 

17.  —;^x{a  +  by. 


14  MULTIPLICATION. 


Case  Ill^Continued.    l*age  41, 


20. 

rt^  —  ah  A-  W' 

a  ■\-  b 

«3       ^^2^  j^  ^^l2 

ci^h       (lb-  +  b^ 

0?                      +  ^^,  Ans, 

21. 

a^      ab  +  Z>2      • 

a^  +  rtJ  -f  h^ 

a^  —  a%  -h  aW 

a%      aW  +  «&3 

aW      ab^  +  Z*4 

«4                ^  ^2^2                _^  ^4^      ^,i^. 

22. 

X^  +  .!•  +    I 

a;2  —  X  -f-  I 

^  _|-  :^;3  _|.  ^2 

—  X^  —  X^  —  X 

--  X^  -}-  X  -\-   I 

x^           +  x^          -\-  I,     Ans. 

23- 

3^2  _  2xy  +  5 

X^  +  2iC^  —  6 

3^:^  —  2.?-3^  +5^^ 

6x^y  4x'^j/^  -\-  loxy 

—  18:^2  +  120;^  —  30 

2fX^  -f  47?y  —  \2,x^  —  /\x^y^  +  22^^  —  30,     Ans, 


24.         ^ax  —  2ay 
6ax  +  ^ay 


24a^x^  —  i2a^xy 

i2a?xy  —  6cfy 

24^^  —  6ahi^,     Ans. 


MULTIPLICATION". 


15 


25- 


29. 


30- 


31' 


d  -i-  hx 
d_-\-  ex 


d'^'+  hdx  +  cdx  -\-  bcx^,     Ans. 


27.     x  +  y 

^  —  y- 

x^  H-  xy 


xy  —  y' 


28.      (V^  +  I 

a  -\-  1 


a^  -\-  I 


a^  —  y%  Ans. 

x^  +  2xy  +  «/2 

^  +^ 

a;^  +  22:2^  +  xy'^ 

x^y  +  2iC?/2  -\-  y^ 


a^  -\-  a^  -\-  a  -\- 1,  ^?i.s. 


x-\-  y  -^  z 
x-Vy  ^  z 

Q?  -\-    xy  -\-    xz 

^y         +y^+  y^ 

xz  -\-    yz  -\-  z- 

x^  +  2xy  +  2XZ  ■}-  y^  +  2yz  +  z^,     Ans. 


Page  45. 

Notes.  —  i.  For  the  details  in  the  following  examples  the  teacher 
is  referred  to  the  formulas  in  the  ])receding  articles  in  the  text. 
(Arts.  101-103.) 

2.  The  learner  should  be  able  to  write  the  following  answers  by 
means  of  the  preceding  formulas. 


16  MULTIPLICATION^. 

1.  (a  +  i)  {a  +  i)  =  «^  -\-  2a  -]-  I,  A71S. 

2.  (2a  +  i)  {2a  +  i)  =  4(^^  +  4«  +  I,  Ans. 

3.  (2a  —  b){2a  —  h)  —  40?  —  4ab  +  Ij%  Ans. 

4.  {x  +  y)  {x  Jr  y)=x^  -{-  2xy  -\-  if,  Ans, 

5.  {x  —  y)  {x  —  y)  ^  x^  —  2xy  +  y%  Ans. 

6.  {1  -\-  x)  (i  —  x)  ^^  1  —  x^,  Ans. 

7.  {iy^  —  2/)  (7^^  —  ^)  =  49/  — 14^^  +  ^^,  ^ns, 

8.  (4m  —  3^)  (4^^?  +  2>^i)  =  i6?m2  _  gn^^  Ans. 

9.  (a;2  —  ij)  {x-  -{-  y)  =zx^  —  y\  A  ns, 

10.  (i  —  72;)  (i  +  72;)  =  I  —  492;^,  Ans. 

11.  (42:  —  i)  (42;  —  i)  =  162;^  —  82:  +  I,  ^?2S. 

12.  (5&  +  i)  (5^  +  i)  =  25^^^  +  loZ^  +  I,  Ans. 

13.  (i  —  a;)  (i  —  2:)  =  I  —  22;  +  a;^,  Ans. 

14.  (i  +  22;)  (i  +  22;)  =  I  +  42.'  +  ^x^,  Ans. 

15.  (8^  —  3«)  (8Z»  —  3«)  =  64^2  —  48«&  +  9^2,  ^?Z5. 

16.  (rtZ>  +  cf/)  («&  4-  cd)  =  aW  +  2ahcd  +  c^t^^,  vl^^s. 

17.  (3«  —  2?/)  (3<^  +  2?/)  =ga^  —  ^if,  Ans. 

18.  {x^  +  ?/)  (.T^  —  y)  =1  x^  —  y%  Ans. 

19.  (2;  —  y'^)  {x  —  y^)  =:x^  —  2xy'^  4-  y\  Ans. 

20.  (2^2  +  a;)  (2^^  —  x)  =  4«^  —  2:^,  A71S. 


PROBLEMS. 

Page 

44, 

Let 

X  —  the  number. 

Then 

22: 

3    ''■ 

Multiplying  by  3, 

2X  —72. 

Dividing  by  2, 

2:  —  36,    A?is. 

Let 

X  —  No.  of  chickens 

Then 

3^:    18. 
4 

Multiplying  by  4. 

3^  —  72. 

Dividing  by  3, 

a;  —  24  chickens,     ^ 

MULTIPLICATION.  17 

Page  43. 

6.  Let  X  =  number. 

Then  ^  _  ^  =  8. 

3        2 

Multiplying  by  6,      4./;  —  3a;  =  48. 

X  ^  48,   y4?2,S'. 

Proof. —  =  32  —  24  =  8. 

3  2 

7.  Let  ^'  =  number  of  army. 

Then  ^  =    840. 

7 

Multiplying  by  7,  s^  =  5^^°- 

Dividing  by  3,  x=  i960,     Ans. 

8.  Let  -  X  =  worth  of  yacht. 
Then                                  ^  =    1^360. 

o 

Multiplying  by  8,  3a;  =  $2880. 

Dividing  by  3,  ic  =    I960,     Ans. 

o.     Given  —  =  20. 

^  5 

Multiplying  by  5,  40;  =  20  x  5. 

Dividing  by  4,  a;  =    5  x  5  =  25,  A)is. 

10.  Given  —  =  20. 

4 

Multiplying  by  4,  5a;  =  20  x  4. 

Dividing  by  5,  x  =    4  x  4  =  ^^>   ^^^•'^• 

11.  Given  —  =  24. 

7 

Multiplying  by  7,  3a;  =  24  x  7 

Dividing  by  3,  :c  =    8  x  7  =  56,   A71S. 

n-  4^  Q 

12.  Given  —  =  26. 

II 

MultipMng  by  11,  4a'  =  28  x  n. 

Dividing  by  4,  rt  =  7  x  n  =  77?   ^^^^' 


18 


MULTIPLICATION. 


13.     Let 

Then 


Multiplying  by  6, 
Dividing  by  5, 


X  =z  number  of  apples. 

KX 

^X  =:    180. 

2;  =  36  apples,    Ans. 


4- 

Let 

X 

number  of  s 

Then 

SX 

7 

30  cows. 

Mult,  and  dividing, 
And 

X  — 

70  -f-  30  _ 

70  sheep; 
100  animals. 

5- 

Let 

-     X 

one  part. 

Then 

SX 

4 

other  part. 

And 

-ix 
^+4    - 

28. 

Multiply 

Uniting 

Dividing 

ing  by  4, 
terms, 

by  7. 

4X  +  s^ 

rx  - 

X 

112. 
112. 
16,) 

And 


Zx 


:=    12. 


A7IS. 


Arts, 


16.     Let 
Then 

And 


X 


X  =  whole  number  of  plums, 


— I —  =  number  given  away, 

X ^  =z  10,  number  left. 

3       4 


Multiplying  by  3  and  4, 

i2x  —  4X  —  T,x  =  120. 
Uniting,  dividing,  etc., 

a;  =  24  plums,    Ans, 


17.     Let 
Then 


Mult,  by  3  and  6, 
Uniting,  dividing, 


X  =  number. 

%A^  JO 

-  -\-        =  21. 

3       6 
6x  +  3.T  =  378. 

X  =  42,     ^/i5. 


DIVISIOK. 


19 


t8.     Let 
Then 


X  =  11  umber, 


19, 


20. 


X  X 

4       6 


Mult,  by  4  and  6,  6a; 

Uniting  and  dividing, 


4:/;  =:    288. 

a:  =:  144, 


Ans* 


Let 
Then 

And 


Mult,  by  3, 
Dividing  by  5, 

And 

Let 
Then 


X  =  one  part, 
■ —  =  the  other. 


12;  4-  —  =  36. 
3 


5rr  =  108. 
^=     2if;| 
22;  ^     >•  Arts, 

7  =  '4*- ) 

X  =  No.  of  bu.  from  one. 

^  =       "         "  the  other. 

7 


And 

X 

,    3^^ 
7 

21. 

Mult. 

by  7, 

lore 

147. 

Dividing  by 

10, 

a;  — 

14A 

bushels;  ) 

Anc. 

3^ 
7 

"10 

< 

■  1 

DIVISION. 

CV?se  J,  Page  47 » 

13- 

7^. 

21. 

C5. 

3- 

2ab. 

14. 

Sd. 

22. 

;26. 

4. 

s^y- 

15- 

61). 

23- 

4^. 

5- 

5- 

16. 

Idf. 

24. 

2a; 

• 

6. 

2  a. 

17. 

gag. 

2/ 

7- 

sab. 

26. 

8^5c2. 

8. 

Sbc. 

C«^ 

^e  If  Pafje  48. 

27. 

Ga-^ 

9- 

47)171. 

19. 

d\ 

28. 

5«^- 

12. 

—  3O' 

20. 

:^. 

« 

29. 

7^Y 

-47^5. 


20 


DIVISION. 


30.  abc. 

31.  2abc, 

32.  ^x^yh. 

34.  i2clH^y. 


35- 


a 


36.     12X^Z^. 

37.   iirri^n. 


Case  II,  I*(ifje  40. 

4.  b^  +  c3  +  d\ 

5-   3^  +  5- 

6.   3^c  —  I  4-  4^'. 


7.    2^?/^ 


2 


2/ 


8.  —  2a;  +  ?/. 

9.  y^  -{-  z—  I. 

10.  — 5«^ — 4b-\-6. 

11.  3^^^  —  3^. 

12.  —  4x^  —  ^cl^ 
+  ax. 


13.  rt^  —  5«  +  2Z». 

14.  I  4-  5«  —  gaJ. 

15.     2^  —  4b  —  5c. 

16.  2  (6«  +  by^ 

+  3.r  (ft  +  ^)^. 

17.  9^?;—  9?/. 

18.  x{b  —  c) 

—  a{b  —  c). 

19.  3ff2  —  2«. 

20.  a  —  a^  -{-  a^. 


Case  III,  Page  51, 

3.  The  dividend  is  the  square  of  the  divisor. 
Hence  the  quotient  i^  x  -\-  y,  Ans. 

4.  The  solution  same  as  Ex.  3.     a  —  b,  Ans. 

5.  The  dividend  is  the  cM^e  of  the  divisor. 
Hence,  a^  —  2ab  -\-  W,    Ans. 

6.  a  -^  b  )  ac  -\-  be  -\-  ad  -\-  bd  {c  -\-  d,     Ans. 

ac  +  be 

ad  +  bd 

7.  a  -\-  b  )  ax  -\-  bx  —  ad  —  bd  (x  —  d,    Ans. 

ax  +  bx 

—  ad  —  bd 

8.  X  -{-  2y  )  2x^  +  ']xy  +  6y^  (  2X  -\-  ^y,     Ans, 

2X^  +  4xy 

Sxy  +  6?/2 

9.  By  Art.  103,     «2  _  j2  _  (^^  _^  j)  (^  _  j)^     and 

(ft  +  b)  {a  —  b)  -r-  (ft  +  Z*)  =  ft  —  J,     ^?zs. 

10.  The  soUition  same  as  Ex.  9.     x  -\-  y,  Ans. 


DIVISION.  21 

II.        a  —  h)  a^  —  h^    {ci^  +  cih  +  H^,    Ans, 


a'b 

a%- 

-al^ 

aW- 

-¥ 

aW  - 

W 

12.     2a  +  3 J  )  6^2  _j-  iT^ab  +  ^W  (  3«  +  2^,     Ans, 
ist  prod.,  6^2  _|_  ^f^^^     2d  div.,  4^^  +  6R 
2d      "       \ab  +  6^*2. 

13-     ^«  —  3  )  rt^  —  rt  —  6  (  «  +  2,     ^y?s. 
ist  prod.,  c^  —  T^a.     2d  div.,  2a  —  6. 
2d      "       2a  —  6. 

14.      ( «^ — 3 w^^'^'  +  3 «!.l'^ — X^^-^{(1 — .?;)  :=  ft^  —  2ax-\-x^i    ^ ^^^« 

Note, — The  dividend  is  the  ciLhe  of  the  divisor. 
Hence  the  quotient  is  the  square  of  the  divisor. 

15'     3^  —  6  )  6:r4  —  96  (  2^3  +  4a;2  -f  8ic  +  16,     ^^25. 
ist  prod.,    dx^ — \2x^.         2d  div.,   \2X^. 
2d      "       \2x^ — 242:^.         3d    "      24^^. 
3d      "       240^2  —  482-.  4th  '^      48a;  —  96. 

4tli    "       48a;  —  96. 

16.  .T  +  2  )  a;2  4-  7^  +  10  {x  -\-  5,     ^?2s. 
ist  prod.,  x^  +  2:?:.         2d  div.,  5a:  +  10. 
2d      "       ^x  -\-  10. 

17.  .T  —  3  )  rc^  —  ^x  +  6  (  .T  —  2,     ^/?s. 

ist  prod.,  x^  —  -^x.        2d  div.,   —  2X  ■\-  6. 

2d      "       —  2a;  -f  6. 

18.  (c^ — 2c:c  +  .^;^)  -T-  (c—x)  =  c— .T,  ^47Z5.     (Art.  102.) 

19.  {a^ -^ 2ab -\- b~)  -^  (a  +  <^)  =  ci  +  b,  Ans.     (xA.rt.  loi.) 

20.  22  («  —  b)  -^  11  {a  —  b)  —  2  {a  —  b),     Ans. 


22  DIVISION". 

PROBLEMS. 

Page  51. 

1.  Let  X  =  son's  age. 
Then                         5x  —  4  =  father's  age. 
And                            6x  —  4  =  56  years. 
Adding  4  to  each  side,     6x  =  60     "- 
Dividing  by  6,  x  =  10  yrs.,  son;      )     . 
And                           ^x  —  4  =  46    "     father.  )  ^ 

2.  Let  X  =  Frank's  number, 
Then                   3a;  =  John's         " 
And                    4X  =  60  marbles. 

Dividing  by  4,     x  =  15,  Frank's  number;  )     . 
And  3:?;  =  45,  John's  "        1 

3.  Let  X  =  one, 
Then                   5:?:  =  other. 
And                    6x  =72. 

And  5a;  =  60.   ) 

4.  Let  a^  =  No.  given  to  one, 
Then                        42;  —  3  =    "       "       "  the  other. 
And                         -5.^—  3  =  57  pears. 

Adding  3  to  each  side,  5^  =  60      " 

And  4X  —  2,  —  45.   ) 

5.  Let  ^  =  No.  ist  had; 
Then                                            22;  =    "     2d    " 

3^  +  4  =    "     3d     " 
And  6:^  +  4  =  190  cents. 

Subtracting  4  from  each  side,     6x  =186     " 

.'.         X  —  3^  cts. , 
And  2a:  =  62    "      )-Ans, 

«  3.7;  +  4    :=    97      " 


DlVISIOi^. 


23 


6.     Let 
TbeD 
And 

And 


8. 


II. 


12, 


Page  52. 

X  ==  number  of  cows, 
^x  ^       "        "  sheep, 
lort'  =z  200. 
ic  =     2o  cows ;  ) 
gx  =  i8o  sheep.  ( 


Ani>. 


Let 

Then 

And 

Subtracting  3, 


And 

Given 
Adding  3, 

Let 
Then 

And 


X  =  less, 
3a;  +  3  =:  greater. 

4^4-  3  =  57- 

4.-^  =  54. 

•*•        X  —  i.^'^?  less .        f    i 

3-*^  -h  3  =  43L  greater,  ) 


2X  -{-  4X  +  X  —  3  =  60. 
IX  =  6^. 
X  ^=    g,     Ans, 

X  =  No.  of  hours  each  travels, 
4X  = 
Sx  = 
'jx  =  35  miles,  the  given  distance. 

X  =    ^  liours,     Ans. 


"■      "  miles  A 

a        a        a        J^ 


a 

a 


10.     Given 


i4«  +7  =  119. 

Subtracting  7  from  each  side,     14a  =  112. 

Given  20^  —  10  =  130. 

Adding  10  to  each  side,  20 J  =  140. 

J  =       7,     ^?2S. 


Let 

Then 

And 

« 


X  =  No.  bought  of  each  fruit, 
3^  =  amount  paid  for  pears, 
4X  =       "  "      "   oranges, 

5x  r^       ''  "      *•'   bananas. 

12a:  =  60  cents. 

X  :^    5,     Ans, 


24  DIVISION^. 

13.     Let  x  =  No.  of  hrs.  it  takes  both  to  empty  it- 

X 

Then        —  =  part  one  will  discharge  ;n  x  honrs, 
20 

And  -  r=    "    other     "  "  " 


<c 


X  X 

—  +  -=   I. 
20  5 


Mult,  and  uniting,  2sx  =  100. 

X  =^  4  hours,     Ans, 


X 

14.  Given  rr  +  -  =    45. 

Multiplying  by  3,       3a;  +  2:  =  135. 
Uniting  terms,  4X  =  135. 

Dividing  by  4,  x  =    ^s^,     Ans, 

15.  Let  X  =  number. 

Then  -  +  3=8. 

2 


Subt.  3  from 

each  side 

1 

'9 

X 

2 

5- 

Multiplying 

by 

2, 

X 

10,       A71S* 

16.     Let 

X 

A's 

number, 

Then 

2X 

B's 

a 

And 

SX  - 

C's 

i( 

a 

6x 

42. 

Dividing  by 

6, 

X 

7, 

A'sn 

umber;  j 

And 

2X 

14, 

B's 

"     h 

(6 

SX 

21, 

C's 

«     ) 

Ans 


17.     If  2r?;  dollars  =  A's  money. 

Then      4X      "       =  B's      " 
And        Sx      "       =  C's       " 

14a;     "       =  amount  of  all,    Am% 


♦ » 


DIVISION".  25 


1 8.     Let  re  =  ist  pai-t, 

Theu  3a;  =  2d      " 

4X  =  3d      " 

And  Sx  =  40. 


19.     Let 

Then 

And 


20.     Let 
Then 

And 


21.     Let 
Then 


^  —    5,  ist  part;  ) 
3:^  =  15,  2d    "       >  ^?^s. 
^x  =  20,  3d    "      J 


X 

-  A's 

number. 

2X 

-  B's 

a 

IX 

-  C's 

a 

6x 

60. 

X 

_  10, 

A's  number;  ) 

2X 

_  20, 

B's       "         yAns, 

IX 

—  30. 

C's       '^         ) 

X 

ist 

part, 

2X 

2d 

(< 

3^ 

3d 

(( 

6x 

—  48. 

X 

8, 

ist  part;  ) 

2X 

16, 

2d     "      >■  An&» 

3^ 

—  24, 

3d      "      ) 

a:  — 

the  number. 

f-== 

23. 

ZX 

18. 

Subtracting  5. 

4 

Multiplying  by  4,     30;  =  1 8  x  4. 

Dividing  by  ^,  a;  =    6  x  4  =  24,    Ans, 


26 


FACTOKING. 


FACTORING 


Page  54, 

5- 

(w  +  2n)  {m  +  27i). 

3- 

2  X  3  X  3««&5. 

6. 

(4a -h  i){4a  +  i). 

4- 

2  X  2  X  s^^'^^yV' 

7. 

(7 +  5)  (7 +  5). 

5- 

5  X  Taaabbcc. 

8. 

(2a  —  sb)  (2a  —  3&). 

6. 

3  X  7^2/^^^^;. 

9- 

(y  +  i)(^+  0- 

7. 

I  ^xxyyyz. 

lO. 

(,   _,2)(i_,2). 

8. 

5  X  ^ahhcxxx. 

II. 

^x"^  _}_  ?^«)  (a;"'  +  y"). 

9- 

7  X  iiaahccd. 

12. 

(20"  —  i)  (2«"  —  i). 

lO. 

5  X  i;^77imnmix. 

13- 

(a'^  +  b^)(a^  +  b^). 

14. 

{ax^  +  ?/)  (ax^  4-  ?/). 

Case  JTJ,  Page  55, 

4.  ^  (2/  +  ^  +  3^)- 

5.  2a{x-\-y  -  2z), 

6.  $bc  [x  —  2X  —  a). 

7.  Sdm  (n  —  3). 

8.  7a  (5m  +  2x), 

9.  2Td  (bx—  2my). 

10.  3«2(2j  +  3c). 

11.  jaxyizx  -+-  5). 

12.  5(5  +  3^'^  — 4a^y)- 

13.  a;(i  +  ^  4-  ^^)- 

14.  3  (a;  +  2  —  32/). 

15.  19^^(0;  —  i). 

Case  III,  Page  56, 

3.  {a  +  b){a-\-b). 

4.  {x  ^y)(x  —  y). 


Case  IV,  Page  57 • 

2.   {a  -\-  x)  {a  —  x). 

3-   (3^  +  42/)  (3^  —  42/)- 

4.  {y  +  2)  (2/  -  2). 

5-   (3  +  ^)  (3  -  ^)- 

6.  (a  +  i)(«  —  i). 

7.  (i  +  J)(i-^). 

8.  (5«  +  4^)  (5«  —  4^). 

9.  (2.?;  +  y)  (2X  -  2/). 

10.  (i  +  4«)  (i  —  4«). 

11.  (5  +  0(5-  i)- 

12.  Oi-2  +  2/')  (;r2  -  2/2). 

(Art.  ] 

13.  {ax  4-  Z»2/)  {ax  —  by). 

14.  (m^  +  n^)  {m^  —  n^). 

15.  (a'"+  ^^'O  (^^'"  —  *")• 


03-) 


FACTORING.  2? 

Case  V,  Page  59, 

2.  7^  —  1  =  {x—  \)  (a;2  -\-  X  -{-  i),     Alls. 

3.  X  —  y)x^  —  y^{x^-\-x^y-\-  oi?y^  -\-  xhf  -\-  xa^  +  ^^ 

^^  —  ^^ 

:i(^y  —  x^y'^ 

a^y"^  —  x^y^ 
x^y^ 
x^y^  —  x^y^ 

x^y^  —  xi^ 

xy^  —  l^ 
xy^  —  y^ 

(x  ^y){^  ■{•  x^y  4-  a^?/^  +  x^y^  +  xy^  -\-  y^),    Ans. 

Note. — As  the  last  term  in  the  dividend  is  not  used  till  the  final 
operation,  there  is  no  need  of  bringing  it  down  every  time. 

4.  x^  —  I  z=  {x  —  i)  (x  -{■  i),     Ans. 

5.  I  —  s^y^  =  (i  —  6?/)  (i  +  6y),    Ans. 

7.  Z>2  —  a;2  =  (5  —  x)  (b  +  x)f    Ans. 

8.  d^  —  :^z=  (d  —  z)  (d^  +  d^z  +  dz^  +  z^),    Ans. 

9.  a^—l^  =  (a— h)  («5 + a^h  +  a%^  +  ct%^ + a¥ + h%  A  ns, 

10.  X— i)x^—  i{a^-\-x^-\-X'\-i 

x^  —  a^ 


3^ 

«3_ 

X^- 

-X 

X  — 

I 

X  — 

I 

tr^  —  I  =  [x  —  \)  i^  -\- x^  •\-  X  -\-  1) ,     Ans, 

Note. — If  the  powers  of  i  were  expressed  in  the  second  factor,  it 
would  take  this  form  :  cc^  +  x^i  +  xi^ +  1^.  But  as  all  powers  of  i  are  i, 
they  may  be  omitted  from  the  literal  terms.     (Art.  94,  Note.) 


38 


DIVISORS. 


11.  I  —  a^=  (i  —a)  (i  -\-  a  -\-  a-  +  a^  -\-  a^  +  a%  Ans. 

12.  a^—i  =  (a—i)(a'^  +  a^-\-a^-\-a^  +  a^  +  a^-\-a-j-i),  Ans. 


14.  x^-{-y^ 

15.  «Hi 

16.  a^-\-i 

17.  1+^^ 

18.  i-{-a^ 

19.  i+b^ 


rage  60. 

(x  -{-y)  {x^— x^y  -\-  xhj'^ — xy^  +  y^),  A  ns. 
{a+i){a?' — a-\-i),  Ans. 
{a-\-i){a^ — a^-\-a^ — «+i)j  Ans. 
(i+?/)(i— 2/  +  ^^),  Ans. 
(i+ft)(i — a^a^ — a^  +  a^),  Ans. 
{iJ^-h){\—b  +  l^—¥  +  ¥—h^-\-h%  Ans. 


DIVISORS. 


Page 

61. 

Page  63. 

Page  66 

3.  X. 

4.  1). 

3.   3«c. 

6. 

X      y. 

5.  ac. 

4.   2«a;^. 

7. 

a  -{-h. 

6.     2X. 

5.  4«3a;V. 

8. 

^+2. 

7.  'jm. 

8.  6al). 

6.  6ax^z^. 

9- 

^  +  Z' 

10. 

a^  —  a  — 

2)  a^  —  :^a  +  2 

«2  _     ^7  —  2 

(I 

Cancelling  — 2,         —  2a  +  4        we  have 

«  —  2  )  a^  —    a  —  2(^4-1 
a^  —  2a 

a  —  2 
a  —  2 

Dividing  the  greater  quantity  by  the  less,  changing  the 
signs  in  the  remainder,  and  cancelling  the  factor  2,  we  have 
a  —  2  for  the  second  divisor.  As  this  is  contained  in  the 
first  divisor  without  a  remainder,  it  is  the  </.  c.  d.  required. 
(Art.  142.)  Ans.  a  —  2. 


DIVISORS.  29 


II.    a^-j-4a^-\-4a-{-^)a^-{-;^a^-\-4a-{-i2  {i 

a^  +  4a^  +  4a-\-   3 

— «2  +  9 

«^ — 9  2d  dividend. 


a  +  s)a^-{-4a^-{-4a-^3{a^  +  a-[-j 

d^-\-4a 
a^  +  ZCt 

«5  +  3 

Changing  signs,  for  second  divisor,  and  cancelling  «  —  3, 
we  have  «  +  3  for  the  last  divisor,  which  is  exactly  con- 
tained in  the  second  dividend.     Hence,  «  +  3,  Ans. 


12.        X?  -\-  1  )  01^  -^^  mx^  +  mx  -f  I  (  I 

3?  +1 


Cancelling  mx,        mx'^  +  mx        we  have 

a;+i)a:^+i  {x^  —  x  '\-  i 
a^  -f-  x^ 


Ans.  X  -\-  1, 


13.        a^  —  i^)a^^^    i^a^^aJy^ 


x^ 

+ 

I 

—  a:2 

— 

X 

+ 

X 

I 

X 

+ 

I 

Cancelling  b'^,  a¥  —  b^        we  have 

4ns.  a  —  b  )  d^  —  b^  {a  -\-  ^ 


30 


MULTIPLES. 


14.   ist  dividend,  a^—a^b-\-   ^ab^—^b^ 

^a^b —     aV^ —  2fi^ 
4a^b — 2oaW  + 1 6b^ 


Canceling     19^^)  19^^ — 19^^ 


2d  divisor, 
2d  quotient, 


a — b 
a— 4b 


Ans.  a  —  5. 


a^ — ^ab-^4b^    1st  divisor. 

(I  -{-4b  ist  quotient, 


a^  —  Sab  +  4b'^    2d  dividend. 

a^ —  ab 


■4ab-\-4b^ 
■4ab-\-4b^ 


15.  [Solution  given  on  next  page.] 


3.  ^ea^b'^cH, 

4.  Sox^yh\ 


MULTIPLES. 
rage  68. 

6.  42oa^b\ 


7.  315:?^^%^ 

8.  84171^71^1/^. 


Page  69. 

12.  d^ — b'^  =  (a  +  b)  (a—b). 
a^—b^=  (a^^ab  +  b''^)  (a—b). 

(d^+ab  +  b^)  {a-\-b)  {a—b)  —  a^ -\-a%—ab^—b\  Atis. 

13.  x^ — i=^(x-\-i){x — i). 

X^-^2X-{-l  =  {x-{-  l)  (x-^l). 

{x^-{-2X-\-l)  (x—i)  ^  X^-{-X^  —  X — I,  Alts. 

14.  The  g,  e.d,  is  2a — i. 

{6a^ — a  —  i)-^(2a — i)=3«-l-i. 
(2«2-|_3<^_2)  X  (3«+  i)  =i  6a^-\-  iia^ — $a — 2,  A71S. 

15.  ^{^-{-m  —  2  ^  (m — i)(m  +  2). 


7)1-^ 


I  =  (m — i'^{ni^-\-m-}-i). 


{m^ —  i)  {m  +  2)  =  ju"^  +  2)n^ — w  —  2,  Aiis. 


DIVISORS. 


31 


a> 

g 

•5 

;-i 

»: 

> 

C 

o 

§ 

« 

o 
.2 

• 

g 

c 

•i-H 

o 

PI 

o 

• 

.2 

•f-i 
> 

o 

> 

13 

C8 

a 

a 

•  I-H 

o 

-4-^ 

'd 

TS 

-a 

'd 

'w 

'O 

C3 

1-4 

H 

M 

N 

N 

M 

CO 

CO 

u 

1 

C<0 

rO 

-iJi 

• 

vO 

oo 

00 

r^ 

lO 

N 

M 

^^ 

m 

;-! 

1-1 

1— 1 

M 

M 

■^ 

M 

QJ 

+ 

-- 

+ 

4- 

+ 

+ 

4- 

»\ 

0) 

^^ 

^ 

« 

5^ 

5^ 

^ 

^ 

« 

M 

r-| 

'3 

fO 

CN 

lO 

^ 

^o 

o 

vO 

CO 

^ 

s 

l-t 

CO 

^ 

vo 

vO 

+ 

+ 

1 

+ 

+ 

H- 

+ 

5^ 
CO 

o 

0^ 

"^^ 

'^^ 

"^ 

=^^ 

"^ 

•^c^ 

+ 

^ 

rt- 

a 

(N 

^ 

sO 

o 

VO 

CO 

l-O 

p* 

CO 

M 

H- 

CJ 

CO 

CO 

c^ 

N 

<X/ 

"^ 

+ 

+ 

1^ 

+ 

+ 

CO 

5^ 

5^ 

CO 

CO 

+ 

00 

f-1 

o 

o 

vc 

00 

\o 

M 

I—' 

CO 

o 

M 

c^ 

o 

1 

N 

OJ 

•r—l 

qn 

-^.^ 

-^ 

1^ 

■^ 
^ 

5^ 

•^ 
^ 

» 

r-^ 
1  ^ 

•rH 

M 

M 

VO 

'^ 

o 

lO 

VO 

( 

ni 

?-i 

o 
o 

\ 

M 

M 

t-H 

o 

tn 

"? 

"C 

^ 

(*^ 

H 

fO 

5^ 

CO 

03 

o 

-5 

a;* 

^ 

00 

00 

o 

U-) 

LO 

CD 

u-> 

lO 

lO 

^ 

.2 

'o 

+ 

+ 

h^ 

s 

c 
.1-* 

1 

^ 

^ 

1 

5^ 

CO 

CO 

,0 

03 

?-; 

^ 

c^. 

« 

^ 

o 

N 

CO 

lO 

lO 

<D 

o 

LO 

OS 

"^ 

M 

o 

h- ( 

I-H 

HM 

+ 

04 

-a 

a 

VO 

+ 

ON 

vn 

o 

CO 

•  I-H 

N 

C< 

O 

« 

M 

¥-^ 

f"^ 

TJ 

>% 

M 

M 

-- 

^ 

-fcj 

r^ 

"^ 

+ 

=1^ 

CO 

1^ 

O 

ON 

03 

lO 

% 

CB 

O 

CO 
M 

o 

M 
1 

00 

M 
1 

00 
+ 

+ 

a  .2 
w  ''3 

+ 

•>* 

CO 

1^ 

CO 

1 

B 

m 

1 

N 

«5   ' 
>> 

M 

1 

s 

CJ 

<^ 

5^ 

^ 

X3 

2 

•rH 

c^ 

CO 

fO 

o 

S 

s 

^ 

^^ 

- 

CO 

-« 

rd 

ni 

ff 

o 

s 

a 

03 

u 

fcdD 

c3 

^, 

S 

'd 

^.^ 

a 

C 

■^"' 

>• 

C4 

N 

F 

•  f-H 

) 

•t^ 

i-t 

'-^ 

-e 

'w 

c3 

X 

^w 

H 

C4 

"p. 

:j3 

w 

> 

• 

§ 

32 


KEDUCTIOK     OF     FRACTIONS. 


REDUCTION     OF     FRACTIONS. 


(Jase  I,  Page  7-t. 
I 


4. 

5. 


sac 
d  ' 


6.    J. 


8. 

9- 
10. 


'jabc^ 

• 

a  —  h 

• 

^  +  y  ^ 

x  —  y 
zy  —  3^ 

7—    a 

2X  —    2Z 


II. 


12. 


X 

O?  —  1/2 


13- 
14. 

15- 


^ 

— 

t 

I 

X 

'  +  f 

a 

+ 

I 

X 

a 

I 

I 

x  +  y 


Case  II f  Page  75. 

2.  a  —  X. 

3-   ^  — 


a 


4.  h  —  c. 

5.  ^  +  c4- 


ic^ 


h—c 


6.  a  —  h. 


7.  ^  + 


2  ah 
b  —  a 


8.  a  ■\-  X  -\- 


a 


9.   32;  +  I  — 


a—x 

zy 


4X 


Case  III,  Page  75» 

4xy  —  b 


y 

4-   5^?  + 


a  —  c 


2b 

I  obcl  -\-  a  —  c 

"^         Yb 

a-  +  2ab  -\-  b^  -{-  2X 


a  -\-  b 


7. 


8. 


2. 


X  -\-   I 

i2ac  —  a  -\-  h 

39^'^  4-  3^ 

i2mx 


3- 

4. 

5- 
6. 


24^2^:^ 

4ff^ 
iMc^  4-  24^6?^ 

a;2  —  ?/2 


dcv^xhf  —  Afbx^y 
3«2  —  2b 


Page  77» 


3'  — 


ah 
ac 

2\C^ 

49« 


(im 


5- 
6. 


ir^  —  ?/2 
x^  —  2xy  +  ?/2 
32^3  (.T  +  y) 
8^2  (^  4-  «/)2* 


keductio:n'    Of    fractions 


33 


Case  V,  Page  7S, 

2  ex  2b(l         CpX 


2dx' 
ac^y 
2c^xy^ 


2dx'     2dx 
2l)xy 


2(?X^ 


5- 
6. 


2c^xy 
2C?  +  2r^^ 


2(?xy 


ihx 


7- 
8. 


Zab  +  3^^'     3^^^  +  3^^ 
x^  —  2xy  4-  ?/^ 

^^  —  11^       ' 
a;2  +  2xy  4-  ^2 

a^  —  y^ 
c?  +  «^      15a  —  3 
3«      '         3«      ' 
^(^a:      2ad      he  +  b 
~hd'    ~bd'    Id 


2Wc—2y^d       2>C'^—T,ad 

^b^c-ir^b^d 
$b^c — ;^b'^d 

2bxy       hz       4az 
2bz  '     2hz''     2bz 

ax  4-  ay      6x  +  6y 

2X  4-2?/'      2X  -|-  2^' 

2X^  4-  2?/'^ 


12. 


10. 


II. 


2a;  4-  2y 

a' 

—  2ax  4- 

:^-2 

d^  —  x^ 

a^ 

4-  2  ax  4- 

2;2 

a^  —  x^ 


Case  VI,  Page  79, 

2.  Least  common  multiple  =  4bcx. 

4b^c^       y  _  bxy 


a  _  2acx      he 

2b  ~  4bcx'     X 


4bcx^     46"       4bcx 


3.  Least  common  multiple  =  :^abc. 

cd 3C%/       2X 2bcx      xy 2fixy 

ah      ^abc^     ^a      $abc^     ac       ^abc 

4.  Least  common  multiple  =:  1 2y. 

a  _  6ay      b  _  4by      c  _  2,cy      x 
2       i2?y'     T,       122/'     4       12?/'    y 


i2y      3       i2y      4       i2y 

Least  common  multiple  =  4b^c. 

4abc^      2cd  _  Scd       x^y  _  bx'^y 


\2X 

\2y 


a^c 


2cd 
4b^c  '     "b^c 


ab        4b'^c  '      b^c       4bh'     4bc       4b^c 

6.    Least  common  multiple  =  24^%. 

3       iSa^c       X         24.?: 


2ab 
yjLG 


i6d^b 
24a^c 


iSa^c 
24d^c' 


X 

a^c 


240^0 ' 


I 

8 


I 

24^2^ 


31  KEDUCTION     OF     FRACTIONS. 

7.  Least  common  multiple  =  2'bc. 

2a       ac       cd 2cd      x^y 2xy 

4b  ~  2bc'     bc~  2bc'    hex      2hc 

8.  Least  common  multiple  =:  a^  —  h\ 

a±h_  _  {a  +  bf      a  —  b  _  {a  —  bf      a^  +  ¥ 
'a  —  b~  a?  —  W'    a+b~  a^-l^'    a^—W 

9.  Least  common  multiple  =  6xy{x  4-  y). 

2  (x  -\-  y)  _  4xy  {x  -\-  y)       a  _  6a  {x  -\-  y) 

3(^  +  y)  ~~  ^^y  i^  +  yV  ^y~  ^^y  i^  +  yY 

ab        abxy 

6{x-\-y)~  6xy{x  +  y)' 

10.  Least  common  multiple  =  a^b^. 

d  ad        X  __  hx 

aW  ~  aW^    c^b  ~  c^W 

11.  Least  common  multiple  =  aU^&d. 

X        Wcdx       m  acdm        y         ab'^y 

ac~ab^^'    ¥c~~(¥M'     '^d'^  otMd 

12.  Least  common  multiple  =  xy^z. 

X  _  xh       a  -{-  b  _  ayz  +  byz      d  _  dy^ 
y^~~  xyh'       xy    ~       xyH      '    xz~  xyh 

13.  Least  common  multiple  =  iia^cx^. 

Ill  -\-  n      4cmx^  +  4cnx^      m  —  n 6acm  —  6acn 

3^2     ~         i2a^cx^       '       2ax^    ~       i2a^cx"      ' 


4CX       1 2a^cx^ 


ADDITION     OF     FRACTIONS.  35 

ADDITION     OF     FRACTIONS. 

Page  SO, 

'iac        iiac       Sac        Kac       27«c       . 

3.     ^ 1 1 h  "   -  =  — —  ,  Ans, 

2xy        2xij        2xy       2xy       2xy 

idxz       indxz       iidxz       Adxz       ^gdxz      . 
Saoc        saoc         ^aoc       ^aoc        sabc 

6.     ^±±Al  +  ^Ji^l^l±±j]i,Ans. 


Page  81. 

8.     Given. 

m      X      11      4S«  +  \2x  -\-  2011      . 
4        5       3  6o 

2X        I        ru       Sax  4-  6  4-  gay      . 

lo. 1 1-  ^  = ^  ,  Ans. 

3        2a        4  i.2a 

a  X  ah  —  ac  -\-  hx  4-  ex      . 

0  -\-  c       0  —  c  W"  —  &■ 

X  4-  y      X  —  y      X  -\-  y  -{-  2X  —  2y       -zx  —  y      . 


13. 
14. 

15- 


2xy  xy  2xy  2xy 

2  ■\r  X      z-^ax  _  2a  -{-  2ax+  z    ^^^ 

y  (^y    ~        (^y        '       ' 

a  ah     _ax  —  ay  +  abx  +  ahy      . 

; 1 — 5 ^ f   A71S. 

X  -\-  y      X  —  y  x^  —  y^ 

cd       2y^       hx  _  ^cd^  +  T,oxy  +  ^hdx^      . 

7,x        d         s  ~  15^^^  ' 


a       2n  4-  d       mil  —  2dn  —  d^      . 

16.     -, -, —  = 7^^ ,  Ans. 

d  3^  z^'i 

*  Fractions  should  be  reduced  to  the  lowest  terms  before  reducmg 
them  to  a  common  denominator.     (Art.  175,  Note.) 


36 


ADDITION     OF     FRACTIONS. 


17. 
18. 

19. 

20. 


ad  —  am  +  dii      am  —  diJ      , 

—  -\ = — -^  = ^ ,  Ans, 

y       —  m.  —  7ny  my 

—  X         —  h         nx  —  mx  —  hy 

H = ,  Ans. 

y         m  —  n  my  —  ny 

—  4       —  16        —  14  +  6  —  16 


+ 


=  —  6,  Ans» 


4a      6c       $711  _  4adx  -\-  6bcx  —  bd^n      . 
o        a        ^x  odx 


Page  81 — Continued. 

h             d           .           hx  -]-  2d      . 
2.     a  ■{ \-  c  +  -=:  a  •\-  c  -{ ,  Ans, 

2  X  2X 


a       —X 
3.     ^  +  Y  + 


+  d  am — at/ — hx-\-hd      . 

-L  .  —  =  ^  H z:    — z. '  ^ 

0        m  —  y  bm  —  by 


ns. 


7      xy  -\-  z  ^  b  —c 


^               xy  +  z       . 
=  2td-{-a-\-b— c ,  Ans. 


5-     5^  +  T  + 


a      —  y  ^   2a  —  by      , 


2b 


ns. 


Page  82. 

^      2a       $bd  +  2«      . 


8. 


a  -{-  b 


—  4y  — ;  -^  '^"« 


X  •}-  y  X  ■\-  y  —  a^      . 

^       "  —  — —^ — ■ — ,  Ans. 


—  a 


a 


a 


10.     3^  +  ?/  4- 


a  —  b       3x^  —  2xy  —  y"^  -{■  (t  —  b 


x  —  y 


x-y 


,  Ans. 


II.     —a-\-si  + 


X 


y  ^x-y-a'^^eab-sb'^  ^  ^4,^^.. 


a  —  b  a  —  b 


12.     2a;  +  2^4- 


a-^b       2X^-{-2Xii—2X  —  2y-\-a-\-b      . 

ziz. i    Jx)lS% 


X—I 


X — I 


SUBTRACTION     OF     FRACTIONS.  37 

SUBTRACTION     OF     FRACTIONS. 

rage  83. 

gabc 

a 


d  —  h       ay  —  dm  -\-  hm      . 


a 

7.     — 

m  y  my 

m  \     yl         m  y  my 

(^  -h  3d      sa  —  2d      ^a  -{-  gd  —  12a  -f  8^/ 

ijd  —  ga      . 

=  — ^~  ,  Ans. 

12 


Pfifje  84:. 

m       \  v     I 


.      h  +  d\       li       li  +  d 
10.     —  — -—     =  — + 


y    I       m  y 

=  bi^^!I^^ ,  Ahs. 
my 


h               h  —  my      . 
II. 711  — -—  ,  Ans, 

y  y 


my 

ISOTE.       m  =  —^  • 

y 

hd  +  cli      . 

=  a  -{ , —  ,  Alls, 

cd 

h       d  —  h       2a  -{-  h      d  —  h 

13.     a -\ = 

2  3  2  3 

6a  -[-  2>h  —  2d  -\-  2I  _  6a  ■\-  ^b  —  2d      . 


38  MULTIPLICATION     OF     FKACTIONS. 

a  c      ad  -[-  ay  —  he  -\-  ex 

d  —  X      d  -\-  y  ~     (b  —  x)  (d  -\-  y) 

ad  -}-  ay  —  be  -\-  ex 

bd  —  dx  -^  by  —  xy^  ^ 

X       'id  2X       xdti  2X-\-yl'u     , 

15.     a —  =:ij ^-^  =  a :il^^   Arts. 

•^  y        2  2y        2y  2y 

X  —  11      a  —  b       x^  —  y"^  —  10a  +  10b      , 
16. ^ ; —  =:  — — ^- ,  Ans. 

10       ^  ■\-  y  10^  +  To^ 

2X3/2  3 

_  (iX—iy-\-zc—2X-\-2y-\-(ia  _  4X—y-\-sc-\-6a 

— ^ —  — — —:- — J     JlTlS* 

0  6 


MULTIPLICATION     OF     FRACTIONS. 

Case  If  Page  85, 

5.  Cancel  the  denominator,    h  +  3^,  A71S, 

6.  Cancelling,    --  x  IS  ==  —  ,  Ans. 

4 
2X — 7,11        ,  ,,       6ex—Qe'u-\-Adx — ^dy    . 

7.     -^  X  (3c  +  2d)  —  ■ -^  -^    ,      . ^^,  ^/2S. 

8.  ay.  —  X  6;^;  =  2aZ>c,  ^?is. 

IX 

a  ■\-  b  Kx       a  -\-  b        . 

x^-  =  — 7—-,  Ans, 


202;  +  2sxy        I        4  +  5^ 

10.  Factor  and  cancel  e. 

a  4-  ab  2c^  4-  2«'^J      . 

X  2ac  =  — y ,  Ans, 

he  -\-  c  b  +  I 

A 

2X  4-  ■? 

11.  — ^^  x'2^  =  Sx^  ■{- I2X,  Ans. 

2  3 

12.  -— —  X =  (22'  4-  3)  (^  —  b) 

5  I 

=  2r^j^  —  2bx  -\-  T,a  ^—  2>b,  Ans. 


MULTIPLICATION     OF     FRACTIONS.  39 

13.  -j X  {d  —  x)  ^  abc,  Ans. 

it  —  X 

^        „.  a  -\-  b  a  -\-  1)      . 

14.  Cancelling  ^x,     x  4a;  = ,  Ans. 

15.  Cancelling  82?  —  2. 

-^^  X  (Sz  —  2)  = -,  Ans, 

40Z  —  10       "^  ^  5 

4,  2B  4 


17.     Cancelling  y  —  i,     3a;  (^  +  i)  =  ^xy  ^-  ^x,  Ans. 


m^  X  4-  z         m^ 

X'^  —  Z^  I  X  —  z 


18.     Cancelling  x  -\-  z,     — ^  x  — ; —  =  ;^ — ":v'  ^'^^* 


Case  II,  Pages  86,  87, 


3- 

6xy. 

4. 

dx 

ay 

5. 

x^  —  y^ 
ifz  4-  \jZ^ 

6. 

— X  ^  — 

ft  +  3:^       8       8«  +  24.-?; 


,  Ans. 


3a;  (ft  +  ???)  c       2i^x 

„      ft  +  ^       c^       d(a  -\-h)       . 

8.     — 5—  X  —  =  — '^ -i  Ans. 

(r  X  ex 

2X  —  y       6x  —  2y  2x  —  y         2  (3a:  —  y) 

4X  y^  —  2xy  ~   2,  Ax         —  y  (2X  —  y) 

XX  —  y       y  —  XX 

=  ^ ^  =  ^- ^ ,  Ans. 

,  —  2xy  2xy 


10  MULTIPLICATION     OF     FRACTIONS. 

12.  Change  the  signs  in  the  multiplicand,  then  factor, 
and  cancel. 

4a  —  2b  2b  —  4a  2  (b  —  2a)  , 

P  —  2ab  ~  2ab  —  b^~  b  (2a  —  b)' 

2  (b  —  2a)       (2a  —  b)  _  b  —  2a      . 
b  (2a  —  b)  6a  ^ab    ' 

13.  Reduce  the   mixed  quantity  to  a    fraction;    then 
cancel  the  x. 

ax  -{-  a  4-  b      ax      a^x  4-  a^  4-  ab      . 

X  -T-  = T J  Ans. 

X  by  by 

2:c       XII  -\-  2X       x(y  4-  2) 

14.  X  -\ =  -^^— ^- —  =     ^^         ^ : 

y         y  y 

^\y  +  2)  ^  (cc  +  ^)  _  («/  +  2)  (:r  +  y) 


y  x^  xy 


xil  4-  2X  4-  y^  4-  211       . 

=    ^       —-^ — ' — -,  Ans. 

xy 

^  X  X  y       X  xy 

x^  —  ?/2       x^  4-  y"^       x^  —  ?/*      , 

—  X  —^  = TT^,  Ans. 

X  xy  x^y 

offi       c(^b  4-  2  a 

16.   Reducinff  to  a  fraction,     a  -\ — ^  — -. ; 

°  ab  ab 

c^b  +  2«2       2ab        ,    ,  . 

ab  a^ 


Case  III,  Page  88. 

abc      dx      abclx      , 

2.  —  X  —  = ,  Ans. 

I       cy       y 

ad      b  -{•  c      abd  -f  acd      . 

3.  —  X =  — —  ,  Ans» 

I  xy  xy  . 


MULTIPLICATION     OF     FRACTIONS.  41 


ax      m  -\-  n      mx  +  ^^^      j 

A.     —  X  = — J  Ans. 

1  4a  4 

a^k       AC  _  4ac  +  4ch       . 

6. ^  X    —  =  — ^-^,  Ans, 

^         y  y 

7.  — ——  X  -^ —  = --^,  Ans, 

I  X  —  1  X  —  I 

8.  Cancel  i  -{-  a. 

I  —  a^         yx  ,  .  . 

X  — - —  =  7a^ui  —  a)=^  IX  —  nax,  Ans, 

I  1  +  a       '    ^  ^       '         '     ' 

9.  Cancel  x  4-  y. 

x^  —  y'^  ac       ac  {x  —  y) acx  —  acy      . 

X  — 7 ; r  —  —  }    A71S, 

I         si^  +  y)  3  3 

ri        1  7         a^  +  ah  36'  T.ac      . 

10.  Cancel  a  +  0. x  —r-^ — 77  =  -— ,  Ans, 

^  I  2{a  -\-b)         2   ' 

a;2  4-  I  2«!ic  2a:(^  -[-  2ax      , 

11.     X  —7 r  — ,  Ans. 

I  3  0'^  — 0  3^'  — 3 

^        ,  ,      2:?;?/  [a  —  b)  4X  Sx^y      , 

12.  Cancel  a  —  o.    — ^^ x  -^, — 7,  = t>  ^^^^• 

1  a^—b^      a  +  b 

r.  1  3^(^  l)  2m  6ff??;  . 

13.  Cancel  x  —  i.    --^ x  -1, = ,  Ans. 

^  I  x^  —  I       a;  +  I 

2ab  +  ^  a^y  2«^a:?/  H-  ^rry      . 

14. x  f-T  = —T^—^  ^^^«- 

I  4fl  +  c>  4a  -\-  b 

15.     Change  the  order  of  the  terms  in  the  denominator, 
and  cancel  i  +  ?^ 

I  —  n^          I  , 
X =  I  —  n,  Ans. 


42  MULTIPLICATION     OF     FRACTIONS. 

EXAMPLES. 

Page  88. 

3 

xc  —  d      %^x       qcx  —  2>dx      . 

4 

2.  Cancel  ^  —  i.  -  x =  33;  (y  +  i),  Ans. 

„        ,    „     x^y  +  2x^      X -\- y       (y-\-2)(x-]-y) 

3.  Cancel  x^.    -^-^ x  — ^  =  \^_^_ls__iju_    ^^ 

•^  xy  x^  xy 

xy 
2ax  _  3«5  _  3^      2)^0  _zc^ 

a  '     ac        c  '     2ab  ~  2^' 

2X      35       3c  . 

I  C  2C> 


%o?      Sy      a      . 

^      %%y      ^a      6' 

2  3 

o^  —  W'  a  X  . 

6,     X  — — T  X  T  =  X,  Ans, 

a  a  +  0      a  —  0 


Page  89. 

m^         X  4-  z         m^         . 
X  — —  = ,  Ans. 


*'     x^  —  z^  I  X  —  z 

8.  ^^—  X  ^-  =  6a^v^,  Ans, 

y        I 

9.  Cancel  x  —  y. 

x+ji  ^  x^-2xy+l^      ^  y)(x-y)  =x^-y\ 
x  —  y  I 

10.     Cancel  ^z  —  2. 

2X  +  y        82;  —  2       2x  -^  y      . 
^-  X = —^ ,  Ans, 


40Z  —  10  I  5 


n 


JJIVISION^     OF     FRACTIONS.  4 

\         X  /       \y      xj  X  xy  x-y 

20^ a^b  -\-  2«2 a^{h  -{-  2)  ^ 

ab  ~       ab       ~~        ah       ^ 

(2«2\       2db       cu^(h-\-2\       2ab        ,  . 

"^ab)  ^  -«^  =  ^J— ^  -«^  ^  ^*  +  4'  ^«*- 

13.  Cancel  c  +  ^  and  2a. 

^^ ^-  X -,  =  2a(c  -\-  a),  Ans. 

2a  c  -^  d  V     '     /J 

14.  Change   the   signs  in   the   multiplier,    and   cancel 
2x  —  y  and  2.     (Art.  166.) 

2X  —  y       2y  —  6x  2x  —  y^      2  (y  —  30;) y — ^x 

4X  2xy  —  y'^  4a;  y  (2X  —  y)         2xy 

2bc        i^  —  bc+2bc       ¥-{-bc       b  (b  +  c) 

15.  0  4-  T = r =  "I =  -^ ') 

J  ^b—c  b—c  b—c  b—c    ' 

,         2bc    b"^  -\-  be  —  2bc  _b'^  —  be  _  b  {b  —  c)  ^ 

"^  V+~c  ~~         b  +  c~~b-\-c~    b  +  c~ ' 

V-^b^cl  ^V'b  +  cJ-     b-c     ^      b-{-V 

=  l^,  Ans» 

X  —  y  ax  X 


DIVISION     OF     FRACTIONS. 
Case  I,  Page  90, 

6X^11  2X         , 

5.     — ^ -^  2>^y  —  — ,  Ans. 

2«2  2a  . 

6. '-  0  =:  —r  J  Ans. 

sac  2>bc 


I         ab\  b      , 

\         c  /  0 


44  DIVISION     OF     FRACTIONS. 


8.     ax  -\ :=  ax  -\-  xy,    and 

(ax  +  xy)  -^x-  = ^  ,  Ans, 

aia  -\-  x)       /     ,     X       (t       . 

c?  —  (?      f  .       a^  -\-  ac  ■\-  &      . 

10.  -J— '-(a  —  c)  = 7—; J  Ans. 

0  -\-  c       ^  0  -\-  c 

1 1.  The  numerator  is  a  square. 

(x  +  yf       .     ,      .       X  -\-  y      . 
~ — -^-  -^  (x  -{-  y)  =  — -^ ,  A  7is. 

12.  — ^ — f  ^(a  +  l)  = f„,  A71S, 

a  —  0  •         a^  —  tr 


lO 


II 


12 


Case  II,  Pages  91,  92» 

3  times. 

5- 

ab      ay      a^y      . 
m       X        cax 

mhx       ax       ax^      . 
6aby      2y      4y^ 

'10?  2  "KX        . 

:^—  X  ■—=  —  f  Ans. 

4  ««/       2y 

xy  2  2  . 

— ^—  X  —  = ,  Ans. 

X  —  I       xy      X  —  I 

a  —  I       ax      c^  —  a      . 

X  —  = ,  Ans. 


X  2 

t^x^y^       2oahx       2X^y      . 

^—^  X =  — -  ,  Ans. 

loab        isxy  3 

1 2  (;?;  4-  y)  2ab  ,      . 

—     z         X  —7^-, — \  —(i,  A  us. 
ab  4  (^  +  //) 

'ic^b^  a  a^h  , 

X  -7—j  = ; — tj  A71S 


a  +  b      (>ab      2a  -\-  zb 


DIVISION     OP     IHACTIONS.  45 


X  +  7/       6bc        3^  +  31/      A 
13.         -Jyt  X  —  =       ,      ",  Arts. 
'^  2b^c^        axy         aocxy 

X  —  a       2d       X  —  a      . 

2X1/         Ay  8z/2 

15.  ,       X  -^-  = ~ ,  Anc. 

^  ■\-  y     3^^     3^^  +.  3^^ 

2X^lj'^  211  ^Xlp  . 

16.  ^  X  — ^  =  —     ,^     ,    ,    ^^ 

rt  +  6       3.^'?/^       3az  +  3152 

a.-^        Z>a;       hx^      . 

17.  — 5X  —  =  -^,Ans. 
'  ax^      ax       ci'^ 

Kb         iSab         b         . 
36aa  loby      4ai/ 


20. 


loby      3^ad_Ady     . 


Case  III,  Page  92 — Continued, 

ex       abchnii      . 

2.  dby  -T-  -7-  = ~ ,  Ans. 

^      dm         ex 

ax      m  4-  n      mx  +  7ix      . 

3.  —  X = ,  Ans. 

^  1  4a  4 

a  4-  X       X       ax  4-  x^      . 

4.  — ■ —  X  —  = ,  A71S, 

I  5c  so 

Kx  —  y       Kx      2Kx^  —  Kxy      . 

^  ^         y  y 

6.         — -L-  X  -^; —  = -^-^  ,  ^^S. 

I  X  4-  1  a-  +  I 

7. X  -~ —  =  3^  —  3«^;  Ans, 

'  I  I  + « 


46 


DIVISION     OF     FRACTIONS. 


lo. 


II. 


12. 


13- 


«  +  I 

a  —  I 

a  —  1 

a  -\-  1 
a  —  h 

X  +  y 

x  —  y  ' 

a-\-h 
x^  —  y^ 

a  —  I 

x^-y 

x-\-y 

a  —  I 

a  -\-b  ' 

a  4-  I       a  —  I 

'  •         

a  —  I    '  a  -{-  L 


a  +  I       rt  +  I 
a  —  I       a  —  I 


a^  +   2a  +   I 

a^  —  2a  +  I 


,  Ans, 


a  —  h      a  -\-  h      c?  —W- 
X  — - —  ^ 

X  -\-  y      X  —  y       0?  —  y'^ 


,   A71S, 


x^ — y^      ^-hy ^^ — xy^-\-x^y—y^ 

I  a — 0  a — 0 


X  4-  y       X  —  1/      x^  —  y^      , 
a—h      a  4-  b~  d^  —  1?' 


X 


y 


EXAMPLES. 


I. 


2. 


6. 


Page  94, 

iKahc        I  5«       . 

-^—  X  -^  =  -^— ,  Ans. 
4xyz        $oc       4xyz 


SSicd  ^  _i_ 


'^%  ^^^^ 


2^y        gxy       24sxy 
13a   2cd        ca 


X 


X 


x  —  y_z^ 


14b        x-Yy 
a  -\-  h  2^xz 


,  Ans. 


42ah 
x^  —  y^ 

2Zxyz  ^ 


A  as. 


36c   K^  —  y)  G 


DIVISION     OF     FRACTIONS. 


47 


8. 


^a          a  —  X          a  . 

X = ,  Ans, 

3  a  ^  X 


a^  —  x^ 


4C^—  Sc       X  -^  y  _  4c(c  —  2)  ^  ^  -\-  y 


X  -\-  y        (^ —  4 


X 


^  +  y     "  {c  ^  2){c—  2) 

AC 


c  -\-  2 


,  Ans, 


(P-  — dx      4  id  +  a;) 
10.     ^ X      ^  ' 


d  (d  —  x)       4  (^  -  '.) 

X 


ac  -{-ax       s{c  —  x)       a  {c  -\-  x)       3  (c  —  r) 

_  4d  {d^  —  x^) 
3a  (c^  —  x^) ' 


II. 


2c2           a  ■}-  G 
X 


2C 


a^  +  c^ 


d^  —  ac  -{-  (P' 


,  Ans. 


4  ia^  —  x^)  a  —  X         4  {a^  —  2ax  -\-  o?) 

12.     ^^^^ '-  X  —7 ^  =^-^ ^,  Ans, 

X  3  (« +  ^)  3^ 


13 


«  —  h 


X 


a  -\-  h 


'     {a  +  l){a-\-b)       d^  —^       a^  -\-  2ah  +  IP 


{a  4-  bf 


,  Ans. 


14 


a;  X  —  1 

X 


a; 


a; 


a;2 — I       a;  +  I       (a;+i)(a;+i)       x^  +  2X-}-i 


,  Ans, 


h(P  +  bed  I  +  a 

15-     — — — —  X 


be  {c  -\-  d)  I  ■\-  a 

X 


X  -\-  ax        b{G  ■{•  d)       X  (i  +  «)        b{c  4-  d) 


=  -,   A71S, 
X 


48  SIMPLE     EQUATIONS. 


SIMPLE     EQUATIONS. 

Payes  97,  9S, 

3.  Given  i  —  c  '\-  x  =  a  —  d 
Transposing,  x  =  a  —  b  +  c  —  d,  Ans. 

4.  Given  x  -{-  ab  —  c  =  a  -\-  b 
Transposing,  x  z=  a  -\-  b  —  ab  +  c,  Ans, 

6.  Given  t,x  -\-  a  —  6  =^  b  —  4  -\-  2X, 
Transposing  and  uniting,        x  z=  2  —  a  -\-  b,  Ans. 

7.  Given  x  —  3-fc=  2X  ^  a  —  b 
Transposing  and  uniting, 

—  X  =^  —b  —  c  +  «+3     ' 
Changing  all  the  signs,     x  ^=  b  -\-  c  —  a  —  3,  A71S. 

Note.  — When  it  is  more  convenient,  the  unknown  quantities  may 
be  transposed  mentally  to  the  second  member  and  the  known  quan- 
tities to  the  first  member.  The  correct  result  mav  then  be  set  down 
in  regular  order,  and  thus  save  the  intermediate  step  of  changing  all 
the  signs. 

8.  Given       2y  -\-  be  —  ad  =  y  -\-  27n  —  8 
Transposing,  y  z=  ad  —  be  -\-  2m  —  8,  Ans. 

9.  Given         3rtZ»  —  y  ■{-  d  =^  —  2?/ +  17 
Transposing,  ^  =  17  —  T^ab  —  d,  Ans, 

ic.     Given  4c^-f  27— 4:^  + rZ  =  2S  —  ^x-{-^bh 

Transposing,    4^6^+27  —  28  +  ^'? — 3M  =  4.^—3:?; 
Uniting,  x  =  ^cd-\-d — 2fih  —  i,  Aris. 

11.  Given  J  4-  c  —  4.T  =  32  +  5  —  ^x  -\-  d 
Transposing  and  uniting,     x  =:  2,2  —  c  -\-  d,  Ans. 

Note  — The  6  when  transposed  becomes  minus,  and  cancels  +h. 
Thus,  &— 6  =  o. 

12.  Given  re  +  4  —  2a;  —  3  ■=  3.^  +  4  +  8  —  ^x 
Transposing  and  uniting,        x  =  11,  Ans. 


0  X  E     U  N  K  N  O  W  K     QUANTITY.  49 


Page  10(K 

3.  Given  ^-+  12  =  ^+  i 
^  5  3 

Multiplying  by  5  X3,       9:^:  +.180  =  20.T  +  15 
Transposing  and  uniting,  ii£  =  165 

X  z^      15,    ^M5. 

4.  Given =  62;  —  66 

36 

Multiplying  by  6,  4a;  —  2;  —  36.?;  —  66  x  6 

Transposing  and  uniting,  33a;  =  66  x  6 

Dividing  by  coefficient,  a;  =    2x6 


a;  r=  12,  Ans. 

3 
10       5 


4.T  +  6 

350- 

—  loa; 

>g^ 

14a; 

344 

a: 

24f, 

J7i5. 

4« 

-5* 

3^ 

X 

d 

r^-  4^        3 

5.     Given  —  +  ^  =  35  —  a; 

Multiplying  by  10, 
Transposing  and  uniting. 
Dividing  by  coefficient, 

7.  Given 

Multiplying  by  dx,    4ad  —  5^f/  =  —  sbx 
Transposing,  ^bx  =  ^bd  —  4ad 

Dividing  by  coefficient,  x  =z , ,  Ans. 

yx  2b  +  c 

8.  Given  ^x  —  ^—  =  a 

5  10 

Mult,  by  10,         30a::  —  14a;  =  10^  —  2b  —  c 
Uniting  the  terms,  16a;  =  10a  —  2b  —  c 

Dividing  by  coef.,  x  = -, ,  Ans. 

9.  Given  —X-] 1-^  —  ^ 

3        4         24 

Multiplying  by  24,     — 24a:;  +  8a;  +  i8a,'  =15 
Uniting,  237  =  15 

Dividing,  a:  =  7^,  Ai?s. 

3 


50  SIMPLE     EQUATION'S. 


XX  (* 

lo.     Given  «  +  J  -f  c  =  -  +  ~  +  «  -I-  Z*  + 

24  5 


Transposing, 
Dividing, 


4^ 

2X  + 

X  -- 

4C 

5 

3^ 

4C  — 

40 

5 

20c 

5 

■4C 

X 

1 6c 

15' 

^?iS 

EXAMPLES. 


Page 

101, 

tjpiven 

X         X 

x-\--  +  ~ 

2          4 

14 

Clearing  of  fractions, 

4X  -\-   2X  -\-  X 

56 

Uniting  terms, 

^x  - 

56 

8,  Ans. 

Given 

X 

7X 

h  40 

10 

Clearing  of  fractions, 

5a;  +  10^  _ 

jx  4-  400 

Transposing, 

^x  = 

400 

•    •                      •//      —~~. 

50,  Ans. 

Given 

AX 

rx 

10  ^    -^ 

Transposing, 

X 

10 

3 

Multiplying  by 

10, 

X 

30,  Ans. 

Given 

''     +6 

X—  2 

8 

Transposing, 

14 

2 

Multiplying  by 

X  —  2, 

14  _ 

20;—  4 

Transposing, 

2a: 

18 

•  •            X  — 

9,  ^l?^5. 

OKE     UN^KXOWX     QUANTITY.  51 

^.  2X  +    I 

5..   Given  X  -\ —  =  10 

Multiplying  by  5,  ^x  +  2X  -\-  i  =  50 

Uniting,  70:  =  49 

X  =    7,  A)is. 

c      1^-  6a:  +  II  ox  —  20 

6.  Given  2X  H ' =  18  + 

5  4 

Clearing  of  frac,    ^ox-\-2dfXArAA  =  360  +  45.T— 145 
Transposing,  19a;  =  171 

X  =  9,  ^?iS. 

/y»  /y»  /v» 

7.  Given  -  H h  -  =  78 

234 

Clearing  of  frac.,   6x-\-4X-\-^x  =  78  x  12 
Uniting  the  terms,  i;^x  =  78  x  12 

a;  =  6  X  12  =  72,  A71S. 

on-  4^  —  6        _         x  —  6 

8.  Given  -^— ^ 8  = \-  s 

6  4  ^ 

Mult,  by  12  (Lc.  m.),  Sx  —  12  =  32:  —  18  +  156 
Transposing,  53;  =  150 

X  =      30,    ^7^5. 

^ .  30;  —  5       4a:  —  8 

9.  Given  ic  4- +  ^^— ^ —  =  1 2 

2  0 

Clearing  of  frac,       6x-\-gx  —  1 5  +  4a; — 8  =z  72 
Uniting  terms,  19a;  =  95 

•  •  X  —    s,    xLTliS* 

n-  -         4a:  +  8 

10.  Given  zx—  16  =.  - — ■ — 

3 

Multiplying  by  3,  6a;  —  48  =  4a;  +  8 

Transp)Osing,  2a;  =  56 

a;  =  28,  Arts. 

^.  2a;  —  8        X  +  -12        X 

11.  Given  1 -^-^  +  -  =  30 

4  2       ^3        ^ 

Clearing  of  fractions, 

6x  —  24  +  6.r  -{-  192  +  4x  =  360 
Uniting  the  terms,  16a;  =  192 

a:  =  12,  A?i8. 


52  SIMPLE     EQUATIOi^S. 


12.  Given  -  +  ^  z=  i6  +  - 

2  0  6 

Multiplying  by  6,  3a;  +  2a;  =  96  +  2; 

Uniting  terms,  4^:  =  96 

a;  =  24,  Ans, 

13.  Given  10  = 1- 


2  36 

Clearing  of  frac,    9^  +  3  —  60  =  ^x  -\-  x  —  i 
Transposing,  42;  =  56 

a;  =  14,  A71S. 

6x       ^x 4X 

10        6  ~  15 


14.     Given  tt  +  ^  =  ^  —  i  A 


Eeducing  to  lowest  terms,   ~ — \-  ^  =z 

5         6        15       15 

Mult,  by  30  (1.  c.  m.),         18a;  +  2^x  =  8^  —  38 

Uniting  terms,  $^x  =  —  38 


Page 

102. 

16. 

Given                       ^f  ■ 

6 

-6-f 

2X 

6 

X 

6  +  ^ 

Uniting  terms. 

2iC 

3 

8 

Multiplying  by  3, 

• 
•   • 

2X 

X 

24 

12,   ^7i5. 

17. 

Given 

Clearing  of  fractions, 

•  • 

4X 

5 
16a; 

X 

7  -= - 

15a;  +  60 

60,  Ans. 

18. 

Given 

?x  - 

-4 

X 

-.  +  ' 

Uniting  and  mult,  by  2 
Transposing, 

42; 
3^ 

a;  +  12 
12 

• 
•  • 

X 

4,      ^?ic<?. 

12 


o^n'e    unknown    quantity.  53 


19.     Given  ^+^-,s^i^+,^i 

4        5  5 


20. 


21, 


22. 


23- 


24. 


Transposing, 

-ix       X 
4+S        '' 

Clearing  of  fractions, 

15a;  -\-  4X        19  X  20 

Or, 

19a;        19x20 

X          20,   A71S. 

Given 

b  -\-  c 
a 

Multiplying  by  a, 

X       al)  -\-  ac,  Ans, 

Given 

ax        , 
—  —  d 
n 

Multiplying  by  n, 

ax       dn 

• 
•  • 

dn       . 
X       —  ,  Ans. 
a 

Given 

ax      hx 

—  +  —  _  c 

2         3 

Clearing  of  fractions, 

ytx  +  2hx        6c 

Factoring, 

(3a  4-  2d)  X        6c 

6c         ,_ 

•  •            X  —            .       7,  JLtlS» 

7,a  4-  2b 

Given  * 

2ax  +  h        ex  4-  d 
a                  c 

Clearing  of  fractions, 

2acx  +  he        acx  +  ad 

Transposing, 

acx  ■     ad  —  he 

ad  —  he      . 

X ,  Ans. 

ae 

Given 

2C        h         ^     1    ^ 

a       X       2       a 

Transposing, 

c       he 
a      X       2 

Clearing  of  fractions,      2ex  -\-  2ah  =  aex 
Transposing,  2ex  —  acx  =  —  2ah 

Changing  signs,  etc.,     {ae  —  le)  x  ^  2ah 


2ah 

X  =z  ,  Ans. 

ac—  zc 


54  SIMPLE     EQUATIONS. 

25.  Given  —  -{-^  -{-  ^  =  ^  ^ 

^  3         5         2         6    '      2 

Clearing  of  frac,  2oa-\-24X-{-4^b  =  25a  +  i8ob 

Transposing,  24a;  =  ^a  -\-  135& 

24 

m'  S^  ,    2a       4b   ,    15c 

26.  Given  ~  H =  —  +  -^ 

2353 

Clearing  of  frac,  4^x-\-2oa  =  24b  +  150^ 

m             •          J  ^-  -J               24^+150^—200^     . 
transposing  and  divid.,    x  ■=  —^ ,  Ans» 

^.  3a  +  a;  6 

27.  Given  ^^ 5  =  - 

XX 

Multiplying  by  a;,        ^a  ■\-  x  —  5:^  ==  6 
Transposing,  4a;  =  3^  —  6 

3a  —  6       . 

4 

28.  Given +  i  =  - 

X  -\-  1  a 

Clearing  of  frac,     ax — a-{-ax-\-a  =  a;  4-  i 
Uniting  terms,     '  2ax  —  ic  =  i 

Factoring,  (2a—  i)iz;  =  i 

2a  —  I 

-~ .  X  X  a 

20.     Given  -  4 =  -  +  « 

a      c  —  a       c 

Clearing  of  fractions, 

c^x  —  acx  4-  acx  =  a^c  —  a^  -\-  «V  —  ah 

Factoring,  c^x  =  a^{c  —  a  -\-  c^  —  ac) 


a^ic—a-\-c^ — ac)      . 
=  — 5^ -~ ,  A7I8, 


X  —  2 


ONE     UNKNOWN     QUANTITY.  55 

or 

30.  Given  x  -\-  h  :=  — -  ^ 

Mult.  \)j  x-\-'b,    x^- -\- 2bx -\- b'^  =  x^ 
Cancelling  and  transp.,    2bx  =  —  b^ 

-        Z_       ^ 

2b  2 

31.  (xiven  X  —  a  := — 

''  X  —  a 

Multiplying  by  x—a,     x^—2ax-\-a^  ^  x^  -{-  a 

Changing  signs,  etc.,  2ax  =  a^  —  a 

a  —  I       . 

X  = ,  A71S. 

2 

rr,  .  ,  XX   4-    Xb 

Iransposmg,  etc., —  =  a;  —  b 

4 

Multiplying  by  4,  3a;  +  3^  =  ^x  —  46 

Uniting  terms,  a;  =  7^,  A71S, 

ry-  ^X  2X  —  S(> 

^^  9  3 

Multiplying  by  9,  ^x  =z  gx  —  6x  -{-  168 

Transposing,  2X  =  168 

x  =  84,  Ans. 


rage  103. 

34.  Given  - — [-  x =    25 

4  2 

Clearing  of  fractions,    3a:  +  4X  —  2X  =:  100 
Uniting  terms,  5a;  =  100 

a:  =    20,  ^?25. 

37  X 

35.  Given  80  =  4.?: ^ 

2        6 

Clearing  of  fractions,      480  —  242;  —  3a:  —  :c 
Uniting  terms,  20a;  =  480 

y%        i»  =    24,  Ans. 


4 

56  SIMPLE     EQUATIONS. 

^.  2a:  +  I  ^  +  3 

36.  Given  =20: 

3  4 

Clearing  of  fractions,      8a:  +  4  =  24a;  —  3a;  —  9 

Transposing,  13a;  =  13 

•*•  X  —    I,    JxtlS' 

37.  Given  10  —  2a;  =  ^^ — ^— ^ ^- 

3  3 

Multiplying  by  3,     30  —  6a;  =  3a;  +  4  —  24  +  36 
Transposing,  92:  =  14 

ic  =  if,  ^^5. 

38.  Given  a;  —  3  =  15  —  ^ 

Uniting  terms,  x  -{ =18 

Multiplpng  by  II,       iia;  +  a;  +  4  =  198 
Or,  12a;  =  194 

a;  =     1 6 J,  A71S, 

^.                                                         a:  +  8       a;  4-  6 
^9.     Given  a;  +  2  =  3a;  -I 

4  3 

Clearing  of  frac.,  i2a;+24  =  36a;4-3^  + 24 — 4.r — 24 
Transposing,  23a;  =  24 

^.  -ix       X  —  4       X  —  10 

40.     Given  ^^  H =  x  —  6 

42  2 

Uniting  terms,  - — f-  3  z=  a:  —  6 

4 

Or,  3^  3=  ,•  _  9 

4 

Multiplying  by  4,  3a:  =  42*  —  36 

Transposing,  x  =  36,  Ans, 


0^'E     UNKNOWN     QUANTITY.  57 

^.  IIX —  I  5X —  II         X —  I 

41.  Given  = 

12  4  10 

Clearing  of  fnic,     55'"^'—  5  =  75^'  —  165  —  6a;  4-  6 

Transposing,  14X  =  154 

.*,        X  =    II,  Ans, 

42.  Given  ' —  =    120 

5        ^o 

Multiplying  by  10,  Sx  —  "jx  =  1200 

X  =  1200,  Aiis. 

^.  2a;  -f-  I 

43.  Given  X  —  20  = 

5 
Multiplying  by  5,      s^  —  100  =  —  2.T  —  i 

Transposing,  jx  =  99 

X  =  14I,  Atis. 

44.  Given = \-  12  —X 

2  3 

Clearing  of  frac.,     gx  —  15  =  S  —  4X  -{-  'j2  —  6x 
Transposing,  19:?;  =  95 

•   •  X    S,    jrLflSt 

/-^J  •  I  ~~~  X  2X  I  —  'iX 

45.  Given  — p 1-  10  = ^— 

6  32 

Clearing  of  fractions,     i  —  a-  +  60  =  42;  —  3  +  9^ 
Transposing,  14a;  =z  64 

X  =    4f,  A71S. 

46.  Given  =  b 

«  +  I       a  —  I 

Clearing  of  frac,    ax — x — ax — x  =  a^  —  b 
Uniting  and  factoring,  —  2x  =  b{a^  —  i) 

Changing  signs  and  dividing,     x  =  -(i — a^),  Ans. 


68 


SIMPLE     EQUATIOifS. 


47.     Given 


X 


2  -\-  X 


a  —  h      a  +  b~  df^  —h^ 
Multiplying  by  a^  —  ¥, 

ax  -\-  hx  —  2a  -\-  2h  —  ax  -\-  bx  =z  c 
Transposing,  2bx  =  2a  —  2b  +  c 

2a  —  2b  -\-  c 


Note.  —  This  answer  may  be  ex- 
pressed in  various  forms,  as ; 


•V 

2b 

X 

— 

a 

7% 

c 

2b' 

or 

X 

2a  -\-  c 

-  h 

or 

a        c  . 

X  =  --,  -\ — 5  —  I,  Ans. 
0       2b 


48.     Given 


Za  -{-X 


X 


Multiplying  by  x,       sa  -\-  x 
Transposing,  4X 


X  = 


6 

5^  +  6 
Za  —  6 
Za  —  6 


,  Ans, 


49.     Given 


Clearing  of  fractions. 
Transposing, 


hx 
2 

2fix  z 

^bx  =  6d 


-,      bx 
a  —  — 

3 
6d  —  2bx 


X  =z 


6d      . 
-=r,  Ans. 

5^ 


50.     Given 


8a  = 


Multiplying  by  i  +  a;,    8a  -\-  Sax 
Transposing,  Sax  +  x 

Factoring,  ( i  +  Sa)  x 


X  = 


I   —  X 

1  +x 

I  —  X 

I       Sa 

I  —Sa 

I       Sa 

I  +  Sa 

A71S. 


OKE      UKKNOWK     QUANTITY. 


59 


51.     Given 


x^  -\-  j^x  -\-  ^        4ab 


X  -\-  2 
Reducing  to  lowest  terms,     x  -\-  2 


Transposing 


16^ 
a 

4 
a 


O' 


ar  = 2,  Ans. 


2.     Let 
Then 

And 


Let 

Then 

And 

Adding, 


4.     Let 
Then 


Adding, 


5.     Let 
Then 
And 


a        a 


coat. 


Ans. 


PROBLEMS. 
rage  105. 

X  =  value  of  vest, 
4X  =z 

Sx  = 

X  =    $S,  vest; 
4X  =  $32,  coat, 

X  =  amount  paid  A, 
2X  =        "  "     B, 

SX="  ''    0. 

6x  =  $9000. 
X  =  I1500,  A  received; 
2X  =  $3000,  B 
^x  =  $4500,  C 

number  of  men. 


2X  = 
22a;  = 


"  boys, 


a 


women. 


A7tS. 


25a;  =2  1000. 
X  =      40  men  ; 
2X  =      80  boys;      }•  Ans, 
2  2X  ■=.    880  women, 

X  z=  distance  one  runs, 

2X  =        '^       the  other  runs. 

SX  =z  120  miles. 

X  =z    40  miles  ;  )    , 

o        ,^       ^  Ans. 

2X  =z      80        '• 


f 


60  .  SIMPLE     EQUATIONS. 

6.     Let  2X  =  number  of  barrels. 

Then  loic  =  cost  of  one  kind. 

^x  =    "      "  the  other  kind. 
Adding,  iSx  ^  '$1200. 

Dividing  by  9,    2X  =z  133!  barrels,  A71S. 


8. 


10. 


II. 


Let 
Then 

Adding, 


iC 


a 


Let 
Then 


X  =  number  ist  receives, 
2X  =         "        2d 
Sx_=        "3d 
Sx  =  g6  pears. 

i?;  =  12  pears,  ist; 
2X  =  24  "  2d; 
SX  =  60      "      3d, 

12a;  =  length  of  post. 

3:^  +  4X  +12   =:    12a:. 


^/^s. 


Uniting  terms, 


SX  =  12. 


a;  ^ 


—  12 


12a;  =  if^  ==  2  8f  feet,  ^?is. 

Let  2oic  =  sum  at  first. 

Then     20a; — ^x—^x  =  I72. 
Uniting  terms,     122:  =  $72. 

X  =    $6. 

20:?:  =   -^120,    A71S, 

Let  a:  =r  B's  share, 

Then  2X  =  A's     " 

3^  =:   C'S         " 

Adding,  6x  =z  I300. 

X  =    $50,  B's  share; 
2.'?;  =  ^100,  A's 
3:?;  =  $150,  C's 

Let  X  =  age  of  wife, 

Then  2X  =    "    "  man. 

And  2a:  +  T  8     :     .t  4-  1 8     : :     3     :     2 

Changing  to  an  equation,     4:r  -|-  36  =z  3,?'  +  54 
Transposing,  .t  =  1 8  years,  wife's  age  ;  )    j    ^ 

2X  =  36      ''      man  s 


a 


i( 


Ans. 


a 


ONE     UNKNOWN     QUANTITY.  01 

12.  Let  X  =z  amount  each  invests. 
Then     x  +  $1260  =  {x  —  $870)  2. 

Or,         X  -\-  ^1260  =^  2X  —  $1740. 
Transposing,        x  =  -S3000,  Ans. 

13.  Let  X  =z  one  number, 
Then                     i^H-  25  =  other    " 

And  2  {2X  +  25)  =  114. 

Dividing  by  2,    22;  +  25  =  57. 
Transposing,  2X  =  32. 

X  ^:z  16,  one  number;  )     . 
a;  +  25  =  41,  other    "         ) 

14.  Let  60a;  =  amount  he  had  at  first. 

Then      6ox — 20:6' — 15^ — 12a; — lox  =  $300. 
Uniting  terms,  ^x  =  I3C0. 

.-.        X  =1  $100. 

60X  =z  $6000,  Ans. 

Page  106, 

15.  Let  X  =  the  number. 
Then                     ^x —  17  =  22. 
Transposing,                 ^x  =  39. 

X  =:z  13,  Ans. 

16.  Let  X  zr:  number  of  days  they  worked. 
Then         gx  +  6x  =  450. 

Or,  15a;  =  450. 

^  =    30  days,  Ans. 

ly.     Let  X  =  No.  of  hours  each  is  on  the  road. 
Then  40.x-  ~  No.  of  miles  one  travels, 

302:  =    "      "      ''     the  other  travels. 
Adding,  jox  =  420  miles. 

X  =z      6  hours. 


4o,r  =  240  miles;  )^^^^._ 

^ox  =180     ''         ) 


02  SIMPLE     EQUATIONS. 


i8. 

Let 

32;        one  part, 

Then 

4X       other  part. 

And 

yx  .     28  inches. 

X          4      " 
'ix       12  inches  ;  )    . 
4X        16      "         ) 

19. 

Let 

X  —  Henry's  money ; 

Then 

^x  .  -  Charles'  money. 

And 

8x        $200. 

X  -    I25,  Henry;   |  ^^^^ 
'jx  —  I175,  Charles,  j 

20.  Let  :c  r=  the  time  past  midniglit. 
Then  x  -\-  ^x  =:  12. 
Clearing  of  fractions,    7a;  +  3a;  =  84. 
Or,  102;  =  84. 

X  =^    8/0  hours, 
or  8  hr.  24  min.  A.  m.,  Ans, 

21.  Let  a;  =  number  of  days. 

Then  —  =  part  A  does  in  i  day, 

20       ^ 

—  =    "      '«       "     X  days. 

20  -^ 

By  conditions, ( 1 =      i. 

•^  20       30       40 

Clearing  of  frac,    6a:*  +  42; -f- 30;  =  120. 
Uniting  terms,  13^=  120. 

a;  =      9^  days,  Ans. 

22.  Let  a;  =  number  of  each. 

Then  loo.r  =  amount  received  for  horses, 

45a;  =       "  "         "  cows, 

t^x  ^       "  "  "   sheep. 

Adding,  150.^  =  I4800. 

a;  =  32,  the  number  of  each,  Ans. 


ONE     UNKNOWN     QUANTITY. 


G3 


23.     Let 
Then 

Adding, 


24. 


2X 

IX 


\ox 

X 
2X 

SX 
Sx 


number  ist  receives. 
"        2d        " 
3d        " 

150  oranges. 


=     15 


30  oranges,  ist; 
75  "  2d; 
45       "        3^, 


Ans, 


Let 
Then 


X  =  price  of  1  chicken. 


SX  = 

6x  =z 


By  conditions,  i2x  = 


And 


a 


18:?;  =z 
102:  = 


a 
a 

a 
a 
a 


"    I  goose. 
"    I  turkey. 

"   4  geese. 
"   3  turkeys. 
"  10  chickens. 


Adding, 


40a;  =:   $10.00. 

X  =  $0.25,  2)r.  of  a  chicken ; 
SX  =  I0.75,      "     "  goose  ; 


Ans. 


6x  =  $1.50, 


ii      a 


turkey, 


25.     Let 


X  =  length  of  fish,  in  inches. 


Then 


Addinor 


4  m.  = 

48  in.  = 

X  . 

-  m.  = 

2 


a 

if 

a 


"  liead, 
"  tail, 

"  body. 


to? 


X 

-  +  52  m.  = 
2       ^ 


X  inches. 


(( 


Mult,  by  2,   a; +104  in.  =    2X 

X  =  104  in.  =  8  ft.  8  in.,  Ans. 


26.     Let 
Then 

Adding, 


X 

20  +  ^ 


one  part, 
other  part. 


20  +   2X 
2X  i 

X  : 

20  +  a;  =    60,  other  "      \ 


100. 
80. 
40.  one  part;  / 


Ans. 


G4 


SIMPLE     EQUATION'S. 


27.     Let 
•Then 

And 


28. 


30. 


X  =  less, 
a  —  X  =^  greater. 


a  —  x 


X 

I 


Clearing  of  fractions,      ad  —  dx  =.  ex. 


Transposing, 
Factoring, 

Dividing, 


ex  +  dx  =  ad. 
(c  -\-  d)  X  =  ad. 
ad 


X  = 


a  — X  =  a  — 


ad 


c  +  d 
ac  -\-  ad  —  ad 


c  -\-  d  c  -\-  d 

ac  +  ad  —  ad  ac 


j-livv*.  ii.v>ixig, 

c  -{•  d             c  +  d 

Less  = 

ad             ^      ,              ac 

: ^ ;        Greater       — -— ,, 

c  -\-  d'                           c  -\-  d' 

Let 

12:?;  —    A's  monev. 

Then 

3^          ^ 

a 

8a:  —  1  "         " 

a 

2X  —  I''         " 

Adding, 

13a;        I1222. 

• 
•  • 

X       $94. 

12a;       In 28,  Ans. 

Page  107, 

Let 

X  —  number  lbs.  of  beef, 

Ans. 


Then 
Hence, 


2X  =r 


a 


"     mutton. 


ii 


Adding, 


25a:  =  cost  of  beef. 
40a:  =     "     "  mutton. 

65:^  z=  $39,  cost  of  both. 

X  =    60  lbs.  beef;       ) 

2a:  =  120   "    mutton,  j 


A71S, 


ONE      UNKNOWN     QUANTITY.  65 


31- 


Let 

X       B's  age, . 

Then 

2X  —  A's    " 

Hence, 

2,x       2X  +  15. 

• 

^  _  15  yrs.,  Bsage;) 

2X  —  30    "     A's    ^'      S 

Let 

X       C's  age, 

Then 

x-\-  s  -  B's   " 

And 

a:  +  8  —  A's   " 

ns. 


Z^' 


The  sum  of  the  ages,  3.^+13  =  110  yrs. 
Transposing,  3^  =    97    " 

X  =  32  J  yrs.,  C  ;  1 

^-f  5  =  37t  "     B;  fAni<. 
a;  H-  8  =  4oi   ''     A,  ) 

33.  Let  X  =  votes  for  defeated  candidate, 
Then         x  -]-  i^o  =z     "       "    successful       " 
Hence,     150  +  2a;  =  2500. 

Transposing,      2X  =z  2350. 

ic  =  1175  votes,  defeated  c.;]. 
And  0:4-150  =  1325      "     successful c,  f 

34.  Let  X  =  number  of  artillery, 
Then                3a;  —  20  =        "        "  cavalry, 

3a;  —  20  +  92  =:        "        "  infantry. 
Hence,     'jx  —  40  4-  92  =  1200. 
Transposing,  ^x  =  1148. 

X  =  164,  artillery;  \ 
33:  —  20  =  472,  cavalry;    -  Ans. 
32; —  20  +  92  =  564,  infantry,  ) 

35.  Let  X  =  B's  share. 
Then               x  +  lioo  =  A's 

X  +  $300  =  C's 
Adding,         3^  +  -^400  =  $2000. 
Transposing,  3./;  =  $^1600. 

X  =  $533i,  B's  share; 

X  +  $100  =  %6^sh  ^'s     "        I  Ans. 

X  +  8300  =  $833^    C's     " 


G6 


SIMPLE     EQUATIONS, 


36.     Let 
Then 
Hence, 


2,x 

share 

of 

one, 

5^ 

a 

a 

the  other. 

2>x 

1150.00. 

X 

$18. 

75- 

3^ 

I56. 

25. 

one ;    ] 

5^ 

$93- 

75. 

other,  )  ^ 

37. 


38. 


39- 


Page  108, 

Let  a;  =  price  of  one  horse. 

Then        %6i6—x  =     "      "  the  other  horse. 

Hence,  5:?;  =  6  x  $616  —  6:?;. 

Transposing,    i  lic  =  6  x  '^616. 

=  6  X I56  =  $336,  price  of  one ;    \    . 
i56i6  -  a;  =  I280,      "     "  other,  j     ^^' 


X  = 


Let 

Then 

And 

Adding, 


X  =  age  of  youngest, 
a;  +  2  =    "     "  next  older, 
ri;  +  4  =    "     "  eldest. 


3^  +  6  =  48. 

2;  =  14  yrs.,  youngest; 
a;  +  2  =r  16    "     next; 
a;  +  4  =  18    "     eldest, 


Suppose  A  and  B  are  the  messengers. 
Let  X  =z  days        B  travels, 

And  X  -{-  s  =     "  A      '* 

Then  652:  =  distance  B  '    " 

And       502;  +  250  =       "        A      " 
Hence,  50:?;  +  250  =  65;?:. 
15a;  =  250. 
X  =:     i6|days,  A?is. 


Ans. 


Proof.         50  x  i6|  +  250  =  1083  J  miles. 


6^  X  16 


=  1083I 


<( 


ONE     UNKNOWN      QUANTITY 


07 


40. 


41. 


42. 


43- 


Let 

X 

number. 

Then 

(^  +  75)  * 

—  250. 

Dividing  by  |, 

^  +  75 

625. 

Transposing, 

X 

-  550, 

Ans, 

Let 

^x       one 

part. 

Then 

^x       other  part. 

Adding, 

2,x  —  48. 

X  :=      6. 

3^  =  18, 


Ans. 


Let 
Then 


12.7;  =r  quantity. 
6rc  4-  4^  +  32^  =  13^  =  «• 


Therefore, 
And 

Let 
Then 

Adding, 


a 

X  =  — 

13 

12a      . 
12a;  = ,  Ans, 

13 

Src  =  A's  acres, 
jx  =  B's    " 

12a;  =  540. 
X  —    45. 
Sx  =1  225  acres,  A's  share;  ) 
yx  =  315     "      B's 


<( 


Ans, 


44.     Let  .^  =  number  of  hours. 

Consider  the  cistern  a  unit,  or  i.     Since  in  i  hr.  i  faucet 
will  empty  J  of  it,  another  -^q,  another  t^,  in  x  hours,  all 

Therefore, 


will  empty  -  H 1 

^  "^   6       10       12 


XXX 

6        10        12 


Clearing  of  fractions, 
Uniting, 


» • 


lox  -{-  6x  -{-  sx  =  60. 

2iX  ^=z    60. 

X  :=^  2^  hours,  4^?'S 


68  SIMPLE     EQUATIONS. 

45.     Let  X  —  I  =  ist  part. 

Then  x  -\-  2  —  2^     " 

And  -  =  3d      '' 

4X  =  4th    '' 


<c 


01 

Adding,       62:  n 1-  i  =  39- 

Transposing,  etc.,    192:  ^  2>'>^'^'^' 
Or,  dividing,  x  =  3x2. 

X  =z    6. 
Then  x~i  =    5,   istpart;\ 

a;  +  2  =    8,  2d     '•      / 

4X  =  24,  4th    '^      / 

46.  Let  X  =L  number. 
Then                6a;  +  12  =  66,  the  sum. 
Transposing,              6x  =  54. 

X  ■=    (),  Ans. 

47.  Let  X  =  whole  Xo.  of  sheep. 
Then                              x  —  7  =  remaiDdei'. 

$94 


a; 


cost  of  one. 


A     -  -94       ^  —  7 

Ana  -^  X =  ^20. 

a;  4 

Clearing  of  frac,    94.T  —  65S  =  80a:. 

Transposing,  14a:  =  658. 

X  =    47  sheep,  A}is. 

48.     Let  4x  =  income. 

Then  3.?  =  A's  expenses  i  yr. 

And  3a;  +  I50  =  B's  expenses  i  year. 

By  conditions,     20X  -i-  $100  =r  15a;  +  6250. 
Transposing,  5a:  =  -Si  50. 

.-.     X  =  $30.  ^a;  =  §120,  A71.S, 


ONE     UNKNOWN     QUANTITY.  69 

raffc  109. 

49.     Let  X  ^  number  of  minutes  required. 

Considering  the  volume  of  the  cistern  unity,  or  i,  it 
follows:  If  tlie  supply  pipe  will  fill  it  in  20  min.,  in  i  min. 

X 

it  will  fill  o^n,  and  in  x  min.,  —  of  it.     In  like  manner,  the 
^^  20 

X 

discharge  pipe  will  empty  ^  of  it  in   i  minute,  and  — 
in  X  minutes. 

Then  •  ^-^=    1, 

15       20 

Multiplying  by  60,        ^x  —  3.T  =  60. 

X  =.  60  min.,  Ans, 

Or,  let  X  =  number  of  hours  required  to  empty  it. 

Since  one  empties  ^  of  the  cistern  while  the  other  fills 
■^  of  it  in  I  minute,  the  former  gains  ^  per  minute  in  the 
discharge.  If  to  gain  -^  requires  1  minute,  to  gain  ^  must 
require  60  times  i  minute,  or  60  minutes. 

^or,  ^    '    i%    '■'     I  min.     :     :i- min. 

ic  =  60  minutes,  A}is. 

50      Let  X  =  number. 

Then  mx  —  nx  =  d. 

Factorinar  and  dividing,     x  = ,  Ans, 

51.     Let  $x  =  No.  leaps  of  Greyhound. 

Then        4X  z=    "      "      Hare  takes  after  G.  starts. 
50  +  4.T  =  whole  distance  H.  goes. 

x\gain,  2  leaps  of  G.  =         3  leaps  of  H. 

Hence,  i  leap     "       =  i  of  3     "        '* 

—  3. 

—  2 

And  3.^  leaps 


Qj.  J       <i  ii  —    3.  «  " 


«       _  9^  «         « 


2 


•yO  Simple    EQUAtioi^g. 

Now  we  have  two  expressions,  in  both  of  which  the  hare's 
leap  is  the  unit  of  measure. 

Therefore,  50  +  40^  =  -— • 

Multiplying  by  2,      100  +  8x  =  gx, 

X  =  100. 
And  s^  =  300  leaps,  Ans. 

Or  thus,  Let     x  =z  No.  of  Greyhound's  leaps. 

Then  —  =    "    Hare's  1.  after  G.  starts. 

3 

But    2  leaps  of  G,  =  3  leaps  of  H. 

Hence,  x  =  —  ''       '' 

2 

By  the  problem.  Hare's  number  exceeds  G.'s  by  50  leaps. 
Therefore,         ^^  _  i?  =  50. 
Reducing,  x  =  300  leaps,  Ans, 


52. 

Let 

ax       one. 

And 

ex       other. 

Then 

ax  —  ex       b. 

Factoring  and 

dividing,     x       • 

And 

ah                \ 

ax       ,  one ;    / 

a  —  c              [ 

he                 \  ^^^' 

ex  = ,  other,  \ 

a  —  c             1 

53. 

Let 

2X  —  weight  of  body  in  lbs. 

Then 

^  +  9  —      "       "  head. 

By  conditions. 

a;  +  18        2X, 

- 

X        18. 

Hence, 

2X  =  :^6  lbs.,  body, 
ic  +  9  __  27    "     head. 
9   "     tail. 

Adding,  72  lbs.,  Ans. 


ONE     tJNKNOWX     QUANTITY.  71 


54.     Let  X  =  number  of  hours  2d  travels, 

Then  .         12  +  a;  =        "        "      -'      ist      '-' 

-^  =  miles  I  st  travels  in  i  hour. 
2 

78  +  ^^  r=      «     «        «      "(12+  x)  hrs. 

2 

—  =      «    2d       "      "  I  hour. 


3 

26a; 


a       i( 


"      "  X  hours. 


And        78  +  -^  =  —  « 

2  3 

6  X  78  +  39a;  =  52.'^. 

Transposing,    13:?;  =  6  x  78. 

a;  =  6  X  6  1=  36  hours ; 

.     ,                  262:  .,  y  Ans, 

And  =  312  miles, 

3 

Or,  more  concisely,        Let  x  =  No.  hrs.  2d  travels, 
Then  *  a;  +  12  =        ''        ist      " 

Hence,      (x-\-i2)—  =  — —  =  dist.  ist  travels. 

^  '2  2 

And  .  ^^=    «     2d       " 

3 

By  the  conditions,  =  — — 

Clearing  of  fractions,      52a;  =  39.T  +  468. 

a;  =    36  hours;  ) 

26a;  .1        r  Ans. 
=  312  miles,   \ 

3  -' 

55.     Let  ic  =  son's  age, 

Then  2x  =  father's  age. 

By  conditions,      2X  —  10  =  3:^  —  30. 

a;  ==  20  yrs.,  son  ;      j     . 
And  *  2a;  =  40    "     father,  ) 


74  SIMPLE     EQUATIONS. 

'65.     Let  1202;  =  income. 

Then  40a;  =  sum  spent  Tor  beard. 


iSx 

a 

a 

"  clotliing. 

12a; 

a 

a 

"  charity. 

By  conditions,  1202^—673;  =  I318. 

Uniting  terms,  S2>^  =  ^3^^- 

X  =      16. 
i2oic  =  r^72o,  Ans, 

66.  Let  12:?;  =  sum. 
Then       6x  —  $30  =  A's  share. 

4x  —  lio  =  B's     " 

30;+    18  =  C's     " 

Adding,  13.1— $32  =12:^. 

X  =  I32. 
I2X  =  $384,  the  sum  divided; 
6a;— $30  ^  $162,  A's  share;  (    . 

4:r  — lio  =  $118,  B's      " 
32;  +    $8  =  $104,  C's     " 

67.  See  Book. 

Prif/e  111. 

6S.     Let  3a;  and  4X  =  the  numbers. 

Then        3:^  +  4     :     4^^  +  4     -     5     '     6 
Changing  to  an  equation,     18:2;  +  24  =  2o.r  +  20. 
Transposing,  2:r  =  4. 

3^  =  6 ; 
4:?;  =  8, 

69.     Let  .      3:?;  =  greater. 

Then  2X  =  less. 

Adding,  50:  =  5760. 

.-.     X  =  1 152. 
3.'^  =  3456,  greater;     2x  =  2304,  less,  Ans. 


Ans. 


70. 


7i. 


72. 


0  X  E     r  N  K  N  0  W  X     QUANTITY. 


75 


Let 

Then 

Aud 

By  conditions, 

Or, 

Transposing, 


Let 
Then 


X 

9  +  •'■ 
9  —  X 

g  -\-  X 

9  +  ^<^ 
.     3^ 


the  rate  of  the  current, 
crew's  rate  down  stream. 


up 


a 


2  (9  —  X). 
18  —  2X. 


X  ^  $  miles  an  hour,  A71S. 


600X  =  length  of  rod 
30a;  = 

202:  =: 
122:  := 

loa;  = 


red, 

orange, 

yellow, 

green, 

blue, 

indigo. 

302  inches. 


a 


u 


i( 


a 


a 


a 


a 


a 


a 


a 


—   302 


a 


Hence,       6oox  —  1472^ 
Uniting  terms,       453^*= 

•  /y»      — .    ■ 

•    •  %Mj     — 

600a;  =  400  in.  =  2>zi  f^v  ^ns. 


302    2. 


¥TT 


=  4  m. 


Let 
Then 


24a;  ^  whole  number  of  kings. 
2)X  =  kings  of  one  name. 
6x  =      "      "  another  name. 

a         ic  ((  <( 


a 


a 


Hence, 


ZX  = 

2X  =: 

24a;  —  19Z  =  5  kings. 

.  «  X    ^—*     X . 


ii 


Hence,  there  were  8,  6,  3,  and  2  kings  of  each  name, 
respectively,  Ans, 


73.     Let  X 

Then  x  -\-  1 

And  x^  -\-  2X  -{-  I  —  0? 

Uniting  terms,  2X 

X  +  I 


u 


one  number, 
other 

15- 

14. 


=    7; 
=     8, 


Alls. 


76  SIMl>LEEQtATlOKS. 

74.     (See  Prol).  51.) 

Let  X  =  No.  of  leaps  of  G. 

Then  80  +  J2:  =        ''  ''     D. 

And  X  leaps  of  G.  =  2X  leaps  of  D. 

By  the  conditions,      80  +  f ;2:  =  2a;. 

Multiplying  by  2,     160  +  3:^  =  ^x. 

ic  =  160. 


ic  :=  240  leaps,  ylws. 


75.  The  steamers  will  meet  in  any  number  of  days  which 
is  a  common  multiple  of  20  and  25,  and  tlie  time  of  their 
first  meeting  in  New  York  will  be  the  I.  c.  ni,  of  these 
numbers,  which  is  100  days. 

Again,  since  the  first  makes  i  trip  in  20  days,  in  100  days 
she  will  make  as  many  trips  as  20  is  contained  times  in  100, 
or  5  trips. 

In  like  manner,  the  second  will  make  4  trips. 

Let  X  =  number  of  miles  ist  sails. 

Then         i  trip     :     5  trips    : :     6000  m.     :     x  m. 

X  =  30000  m.,  ist  sails;   \ 

AX  ( 

And  —  =  24000   "     2d     "      }  Ans, 

5  I 

Time  of  meeting    =      100  days,  7 

Or  thus:  Since  the  ist  makes  a  trip  5  days  sooner  than 
the  2d,  it  is  plain  she  will  make  5  trips  while  the  other 
makes  4  ;  now  4  times  25  days  =100  days. 

Again,  the  second  will  sail  f  as  many  miles  as  the  i  st  in 
the  same  time. 

Let  X  =  number  of  miles  ist  sails. 

And  ^x  =       "  "  2d      " 

Then  ^x  =  6000  miles. 

X  =  30000  m.,  ist  sails;     ) 
And  fx  =  24000   '•     2d      "        \  Ans. 

Time  of  meeting  =  (20  x  5)  or  (25  x  4)  =  100  da.  / 


TWO     U  is  KNOWN     QUANTITIES.  71 

SIMULTANEOUS     EQUATIONS. 
TWO    UNKNOWN    QUANTITIES. 

Page  114:. 

2.  Given  x -\- y  =^  12  (i) 
And  a;  —  1/  +  4  =  8  (2) 
From  (i),                                   x=i2—y            (3) 

"      (2),  x=    4^y  (4) 

Equating  (3)  and  (4),     12  —  y  =    4  -{-  y 
Transposing,  2?/  =  8 

Substituting  4  for  ?/  in  (i),      x  ==z  S,  ) 

3.  Given  ^x  -{-  2y  =  48  (i) 
And                                2x  —  ^y  —    6                     (2) 

From  (i),  2:  =  4^1:1^1  (3) 

"      (3),  .^^  (4) 


Equating  (3)  and  (4), 


2 
48  — 2y  _  6  +  3^ 
3  2 

Clearing  of  fractions,     96  —  4y  =  18+97/ 
Transposing,  13^  =  78 

Substituting  value  of  «/ in  (2),  a;  =:  12,   ) 

Given  a:  +  y  ~  20  (i) 

And  2X  -{-  ^y  =  42  (2) 

From  (i),  x=2o  —  y  (3) 

«   (2),  ^  =  ^~^      (4) 

Equating  (3)  and  (4),      20  —  y  =  ^^—'-^^ 
Multiplying  by  2,  40  —  2?/  =r  42  —  3?/ 

Substituting  2  in  (i),  a:  —  i8,  j      ^^* 


78  SIMPLE  EQUATIONS. 

5.     Given                              4X  -\-  ly  —  13  (i) 

And                                3X  -^  2y  =z    9  (2) 

From  (i),                                  X  =  i^-^M  (3) 


9  —  ^y 

3 

13  —  3y  _  9  —_^y 


"    (3),  ^  =  -T^.        W 


Equating  (3)  and  (4), 

Clearing  of  fractions,    39  —  9?/  =  36  —  Sy 

2/  =  3  j  I  ^1^5. 
Substituting  3  for  ?/  in  (2),      2:  =  i,  f 


6.     Given                              3^'  +  2^  —  ^^^ 

(I) 

And                                  X  +  sy  —  191 

(2) 

118  —  2W 

From  (i),                                   X  -         ^     ^ 

(3) 

"    (2),                        ^  —  191     sy 

(4) 

118  —  2«/ 

Equating  (3)  and  (4),    191       52/  —  "       ^ 

Multiplying  by  3,       573  —  15?/  =  118  —  2U 

Transposing,  13?/  =  455 

••.        ^  =    35  ; )  ^^^^^ 
Substituting  35  for  y  m  {2),    x  =     16,  ) 

Given  4X  -{-  sy  —  22  (i) 

And  7a;  +  3?/  =  27  (2) 

22  —  Ky  ,  . 

From(i),  x  = ^^-^  (3) 

"      (a).  ^  =  -^^  (4) 

Equatnig  (3)  and  (4), = 

Clearing  of  frac.,        154  —  35//  =  108  —  12?/  _ 
Transposing,  23?/  =:  46 

y  =    2  ;  )     . 

Substituting  2  for  ?/  m  (3),      a;  :^    3,  ) 


TWO     UJS'KJSTOVVK     QUAXTiTIES.  79 

Case  II f  Page  111 — Continued. 

9.     Given  rr  +  3//  =  19  (i) 

And  ^x  —  2y  =  10  (2) 

From  (i),  X  =19-3?/  (3) 

Substituting  value  oix  in  (2), 

5(19-3^)  —  2y=io  (4) 

Keducing,  95  —  157/  —  2?/  =  10 

Transposing,  etc.,  17?/  =  85 

2/  =    5  '  I  ^ 
Substituting  5  for  y  in  (3),      a;  =    4,  f 

10.     Given  -  +  ^  =    7  (i) 

23  ^  ^ 

And  -  +  ^  zrr     8  (2) 

Mult,  (i)  by  6,  2>x-^2y  —  ^2  (3) 

Mult.  (2)  by  6,  2.T  +  3^  =  48  (4) 

From  (3),  a;  =  ^^^—  (5) 

Substituting  value  of  .r  in  (4), 

'-^^  +  3.  =  48  (6) 

Mult.  (6)  by  3,     84  —  4?/  +  9?/  =  144 
Uniting  terms,  5?/  =  60 

y  =12'/ 

Substituting  12  for  y  in  (5),    a:  =     6, 

IT. 


^ns. 


Given 

2^  +  3?/  —  28 

(0 

And 

3a:  +   27/   _   27 

(2) 

From  (i), 

^        28  -  31/ 

2 

(3) 

Subst.  in  (2), 

84- 
2 

97/ 

— -  +  2y        27 

Mult,  by  2, 

84- 

-  9i/  +  4^  -  54 

Uuitiug  terms, 

5^  -  30 

ci__i  -  J  •  J  _^  J  •      _-   ^    c 

>                           • 

?/        6  ;  I 

80  SIMPLE     EQUATIONS. 

12.  Given  4X  -j-  y  =  4$  (i) 
And  52:  H-  2?/  =  56  (2) 
From  (i),  y  =  4s^4X  (3) 
Siibst.  in  (2),  5a?  -|-  86  —  82:  =  56 

Uniting  terms,  $x  =z  30 

cc  =  I  o ;  I 
Substituting  10  for  x  in  (3),    ?/  =:    3,  f 

13.  Given  ^x  -\-  S  =  yy  (i) 

And  5?/  +  32  :=  7.-1;  (2) 

From  (i),  y  =  ^^  (3) 

Subst.  m  (2),       -^ — -!— ^  +  32  r=  yrc 

7 

Mult,  by  7,       25:?;  +  40  4-  224  =  49a; 
Transposing,  24:?:  =  264 

Substituting  1 1  for  x  in  (3),     y  =    g,   ) 


14.     Given                             4.?;  +  5?/  —  22 

(i) 

And                                7^  +  32/27 

(2) 

^         /  X                                              22  —  t;?/ 

From  (i),                                   a;       ^ 

^  ^                                                   4 

(3) 

Substituting  value  of  x  in  (2), 

154      352/  ,    ,„       „ 

4 
Clearing  of  fractions, 

154  —  352/  +  12?/  =  108 
Transposing,  232/  =r  46 

?/  =  2  ;  ) 
Substituting  2  for  ?/  in  (3),      a:  =  3,  j  ^ 


TWO     UNK]S"0\VN     QLAM'ITIES.  81 

Case  Illf  Page  116, 

17.     Given  sx  +  4?/  =  29  (i) 

And  7x  +  iiy  =  76  (2) 

Multiplying  (i)  by  7,  21X  -\-  28^  =  203 


(( 


(2)  by  3,  21a; +  33^  =  228 


Subtracting, 

5y  - 

25 

Substituting  5  for  y  in 

y  - 

(l),                      X- 

^ '  I  Ans. 
3.  i 

18. 

Given 

9^  —  Ay  — 

8         (I) 

And 

13^  +  iy  - 

loi          (2) 

Multiplying  (i)  by  7, 

6^x  —  28y 

56 

Multiplying  (2)  by  4, 
Adding, 

S2X  +  2Sy  — 

404 

iiSX 

460 

Substituting  4  for  x  in 

X 

(I),          y 

"^ '  i  Ans. 

7,   ) 

19. 

Given 

z^-iy  - 

7           (i) 

And 

\2x-\-  sy  — 

94            (2) 

Multiplying  (i)  by  4, 
Subtracting, 

i2x  —  2Sy  = 

28            (3) 

33y  - 

66 

Substituting  2  for  y  in 

y  - 

(l),                      X  - 

'  [  Ans. 
7,  S 

20. 

Given 

SX+  2y  — 

118         (i) 

And 

x  +  sy  - 

191          (2) 

Multiplying  (2)  by  3, 

3^  +  15//  — 

573         (3) 

Bringing  down  (i). 

ZX-\-    2y  - 

118 

Subtracting,  i32/  =  455 

^  =    35  j  [  j^^^g^ 
Substituting  35  for  2/  in  (i),  2;  =     16.  ) 


82  SIMPLE     EQUATIONS. 


ti.     Given                                      4^  +  5?/ 

(i) 

And                                          ^x  +  2>y 

27        (2) 

Multiplying  (i)  by  3,          122^  + 

15^  - 

66 

Multiplying  (2)  by  5,         35.6-  -f 

1M_~ 

135 

Subtracting,                        23a; 

— 

69 

• 
•  • 

Substituting  3  for  x  in  (i), 

X 

y  - 

2  '  [  ^^^^• 

EXAM  PLES. 

I.     Given                                     2x  -\-  7,y  .  . 

23            (i) 

And                                        52;  — 

-2y  - 

10            (2) 

Multij^lying  (i)  by  2,              4^  +  6?/  — 

46 

(2)  by  3,            15a;  - 

.  61, 

30 

Adding,                                 190; 

76 

• 
•  • 

Substituting  4  for  x  in  (2), 

X 

y  - 

5.   ) 

2.     Given                                     4%  + 

y  - 

34            (I) 

And                                       4y  -\- 

X 

16            (2) 

Multiplying  (2)  by  4,           42;  + 

i6ij  — 

64            (3) 

Subtracting  (1)  from  (3), 

^by  - 

30 

Substituting  2  for  ?/  in  (2), 


2'=  :'U„.. 

:c  :=     8,   ) 


Given 

32  +  4?/  _  27           (i) 

And 

SX  +  sy  -  34        (2) 

Multiplying  (i)  by  3, 

9^  +  12?/  _  81  •     (3) 

(2)  by  4, 

202:  +121/      136 

Subtracting, 

11^         -  55 

Substituting  5  for  x  in 

(3),            y  -     3>) 

TWO     UNKNOWN     QUANTITIES.  83 


Given 

2X-^  VJ  --^  34 

(I) 

And 

5:/;  +  9^  _  5  I 

(2) 

Multiplying  (i)  by  5, 

lo-''  +  zsy  —  170 

Multi]jlyii]g  (2)  by  2, 

lox  +  18^        102 

Subtracting, 

ny       68 

Substituting  4  for  y  in  {i),  a:  =  3,  f  ^ 

5.  Given  ^x  +  7y  =  43  (i) 
And  iia;  +  97/  =  69  (2) 
Multiplying  (i)  by  11,       55X  +  77?/  =  473 

(2)  Ijy  5.  55-^-  +  45^  ^  345 

Subtracting,  32?/  =  128 

Substituting  4  for  2/  in  (i),  x  =  3,  j 

6.  Given  8a;  — 21?/=  t^t,  (i) 
And  6:?;+  35?/  =  177  (2) 
Multiplying  (i)  by  3,        24X  —    63?/  =    99 

"  (2)  by  4,       24:^  +  140?/  =  708 

Subtracting,  2037/  =  609 

y  =    ^'•'^  Ans 
Substituting  3  for  ?/  in  (i),  a;  =  12,  f 

7.  Given  21?/  -f  20.T  =  165  (i) 
And  77?/  —  3o.f  =  295  (2) 
Multiplying  (i)  by  3,          63^  +  6o.i-  =    495 

(2)  by  2,        154^  —  602'=    590 


i( 


Adding,  217^  =:  1085 

?/  =  5  ;  ) 
Substituting  5  for  ?/  in  (i),  a?  =  3,  f  " 


84  SIMPLE     EQUATIONS. 


8.     Given                                    iiic  — 

loij  _  14             (l) 

And                                        sx  -\- 

iy  —  41        (2) 

Multiplying  (i)  by  7,          77^  — 

70?/  _  98 

"            (2)  by  10,       50X  +  70?/       410 

Adding,                             1272: 

508 

• 
•  • 

Substituting  4  for  a;  in  (i), 

y  -  3^  ) 

Page  117. 

Given  6y  —    2x  =  208         (i) 

And  —  4?/  +  iojC  =  156          (2) 

Multiplying  (i)  by  5,  30?/  —  lor?;  =:  1040 

Adding,  26?/             =1196 

Substituting  46  for  ?/  in  (2),  :r  =  34,  f 


10. 

Given 

4^  +  3^  - 

(i) 

And 

5^       7^  - 

6           (2) 

Multiplying  (1)  by  5, 

20X  +  15^  — 

no 

(2)  by  4, 

202;  —  28?/ 

24 

Subtracting, 

43^  - 

86 

Substituting  2  for  y  in 

y  - 

(2),                  X  = 

'  [  Ans, 

4,  ) 

IT. 

Given 

3^-sy  - 

13        (0 

And 

2x  ^  ry  - 

81       (2) 

Multiplying  (i)  by  2, 

6x  —  10?/ 

26 

(2)  by  3, 
Subtracting, 

6X  -\-    211/ 

243 

3Ky  - 

217 

Substituting  7  for  ?/  in 

y  - 

(0,                 X  = 

.6; }  •^-- 

TWO      LNKNOWK     QUANTITIES.  85 


12.     Given 

5^      iy  - 

ZZ 

(I) 

And 

iix  +  i2y 

100 

(2) 

Multiplying  (i)  by  ii, 

ss^-iry  - 

Z^l 

(2)  by  5, 

SSX  +  6oy  - 

500 

Subtracting, 

137^  - 

137 

y  - 

Substituting  i  for  y  in  (i),               x 

8,'  }  ""'''' 

13.     Given 

^  ,y 

5^6 

18 

(I) 

And 

X      y 
2       4 

21 

(^) 

Multiplying  (i)  by  30, 

6:^^+5^- 

540 

(3) 

(2)  by  4, 

2X        y  - 

84 

(4) 

(4)  by  3, 

6x     2>y  — 

252 

(5) 

Subtracting  (5)  from  (3) 

^y 

288 

Substituting  36  for  y  in 

y  - 

(4),               X  — 

36;) 

60,   f 

Ans. 

14.     Given 

1 6a7  +  lyy 

500 

(I) 

And 

ijx—    sy  — 

no 

(2) 

Multiplying  (i)  by  3, 

482;  +  51?/  _ 

1500 

"            (2)  by  17, 

289a:  — 51?/ 

1870 

Adding, 

337-^              — 

3370 

Substituting  10  for  re  in 

(2),          y  - 

10;  ) 
20,   j 

Ans, 

15.     Given 

Sx  +    y  — 

42 

0) 

And 

2X  +  4//   — 

18 

(2) 

Multiplying  (2)  by  4, 

%x  +  i6z/  — 

72 

(3) 

Subtracting  (i)  from  (3) 

15//  - 

30 

y  - 

Substituting  2  tor  y  in  (2),                x 

5.  ( 

ns. 

86  SIMPLE     EQUATlOI^rS. 


i6. 

Given 

2X'  +  4^  — 

20 

(0 

And 

4^'  +  5^  - 

28 

(2) 

Multiplying  (i)  by  2, 
Subtracting, 

4a;  +  8?/ 

40 

(3) 

3!/  - 

12 

Substituting  4  for  y  in 

(0, 

^  - 

X 

2.     j 

ns. 

17- 

Given 

A-^  +  3«/  - 

50 

(■) 

And 

3^       3.?/  — 

6 

(^) 

Adding, 

7^             _ 

56 

Substituting  8  for  x  in 

(2), 

X 

y  - 

8;l  4 

??5. 

i8. 

Given 

z^  +  sy  - 

57 

(I) 

. 

And 

5^  +  32/  — 

47 

(2) 

Multiplying  (r)  by  5, 

I 

SX  +  25?/  _ 

285 

(2)  by  3, 
Subtracting, 

I 

5^  +  9y  = 

141 

i6y  _ 

144 

Substituting  9  for  y  in 

(I), 

«/- 

A,   ) 

/zs. 

19. 

Given 

^  ,y 
2     3 

7 

(■) 

And 

-  H-  - 
3       4 

5 

(2) 

Multiplying  (i)  by  2, 

211 

14 

(3) 

(2)  by  3, 

^  +  ?- 

15 

(4) 

Subtracting  (3)  from  (4), 

3?/       2,?/ 
4          3 

I 

Clearing  of  fractions, 

9y-^  = 

12     • 

Substituting!:  12  for  y  in  (3) 

y  - 

X 

6,    )  ' 

ins. 

TWO     UNKNOWN     QUANTITIES.  87 


20.     Given 
And 


2X  -{-  y  —  so 

X      y 

-  4-  -  —     ^ 
67           ^ 

(I) 

(2) 

nx 

6+y    ^5 

'3) 

^x 

2X           ^              15 

12X  —  7.6'           90 

X        18:  ) 

A 

Multiplying  (2)  by  7, 

Subtracting  (3)  from  (i), 

Clearing  of  fractions, 

.T  =r  18 ;  ) 
Substituting  1 8  for  .r  in  (i),  ^  =  14,  j 


PROBLEMS. 

1.  Let  X  =z  one  number, 

y  =  other     " 
Then  x  +  y  =70  (i) 

And  X  —  y  =  16  (2) 

Adding  2a;  =  S6 

Substituting  value  of  a:  in  (i),      ?/  =  27,  ) 

2.  Let  ic  =  price  of  a  lemon, 
And  ?/  =  "  "  an  orange. 
Then  Sx  -\-  4y  =  56  cents.  (i) 
And  ^x  +  Sy  =  60  "  (2) 
Mult,  (i)  by  2,  16a;  +  8^  =  112  "  (3) 
Subt.  (2)  from  (3),  13^            =52  cents. 

a;  =  4  cts.,  lem. ;  ) 
Substituting  4  for  a:  in  (i),   y  =  6    '-    or'ge,  f  ^ 

3.  Let  .T  =  votes  cast  for  one. 
And  y  —      "       "     "   the  other. 
Then  ^  +  ^  =:z  375                               (i) 
And  X  —  y  =    gi                              (2) 
Adding,  2X          —  466 

X  =  233,  V.  for  one;    | 
Subst.  value  a;  in  (i),    y  —  142,     "      other,  \ 


88                                   SIMPLE  EQUATIONS. 

4.     Let  X  =  greater  part, 

And  y  =  the  less. 

Then  a;  -f-  ?/  =  75                       (i) 

And  ^x  —  ^y—  15                      (2) 

Multiplying  (i)  by  3,  z^  +  ZV  =  225                    (3) 

Subt.  (2)  from  (3),  loy  =  210 


y  =z  21,  less; 
Subst.  value  of  «/  in  (i),         a;  =  54,  greater 


.! 


Ans, 


5.  Let  X  =  price  of  a  horse, 
And  y  =z     ^^        "   cow. 

Then  gx -^  yy  =  1 1200                         (i) 

And  6a;+ 13?/ =r  I1200                          (2) 

Mult,  (i)  by  2,  i8i?;+i4?/ =  $2400                           (3) 

«      (2)  by  3,  i8:^;  +  39^  ^  $3600  (4) 

Subt.  (3)  from  (4),     25?/  =  I1200 

y  =  $48,  pr.  of  cow;    |  ^^^^ 
Subst.  value  y  in  (i),    x  =  I96,       "    horse,  ) 

6.  Let  X  =  No.  of  gentlemen, 
And                               y  =        "    ladies. 

Then  «/  —  1 5  =  ladies  who  remained, 

a;  —  45  =  gent.      "  " 

And  X  =  2  (?/  —  15)  (i) 

5(^-45)  =  2/- 15  (2) 

From  (i),  X  =z  2y  —  ^o  (3) 

"     (2),         s^  —  22s  =  y  -  15  (4) 

Substituting  value  of  x  in  (4), 

5  i^y—  30)— y  =  210 
Transposing,  gy  =  360 

^  =  40  ladies ;         )  ^^^^^ 
Subst.  40  for  y  in  (3),   x  =  50  gentlemen,  f 


TWO      L.NKNOWN     QUANTITIES.  89 


7. 


Page  118. 

Let                                                     X  —  ist, 

And                                                   y       2d. 

Then                                       5  a;  +  2?/        19 

(>) 

And                                          Tx  —  6y          9 

(2) 

Multiplying  (r)  by  3,           15a;  +  6?/  r=  57 

Adding,                                  22a;                 66 

Substituting  3  for  a;  in  (2),  ?/  =  2,  f 

8.  Let  X  =  No.  of  men  in  one  army, 
And  .  2/  =:  "  "  the  other. 
Then  re  +  ?/  =  21110  (i) 
And  20;  +  3?/  =  52219  (2) 
Mult.  (i)by2,  2X  -\-  2y  ^^  42220 

Subtracting,  y  —    9999,  one  army;  )  ^^^^^ 

Substituting  in  (i),     x  =  11 1 11,  other  "       ) 

9.  Let  x  =  digit  in  tens'  place, 
And                                         y  =     "       "  units'    ^• 
Then                                x  -\-  y  =  11  (i) 

And  iz^  4-  13  =  3^  (2) 

From  (i),  X  —  11  —y  (3) 

Substituting  value  of  x  in  (2), 

II  — ?/+  13  =  3^ 
Transposing,  4?/  =  24 

?/  =    6,  units'  digit. 
Subst.  6  for  «/  in  (3),  a;  =    5,  tens'      '' 

The  number  =  56,  Ans. 

10.     Let        cc  =:  A's  share,     and    y  =  B's  share. 

Then  x+    y  =    8570  (i) 

And  sx  +  sy  =  ^2350  (2) 

Mult,  (i)  by  3,    3a;  +  3j!/  =  81710 
Subtracting,  2?/  =    $640 

2/  =  ^320,  B's  share;  )  ^^^.^ 
Subst.  320  for  y  in  (i),  a;  =  $250,  A's     "       f 


DO  SIMPLE     EQUATIONS. 


II.     Let  -  =  the  fraction. 

Then =  -,     or    3Z  +  3  =  ?/  (i) 

y        3 

And z^  - ,     or    4X  =  1/  +  I  (2) 

y  +  1        4 
Substituting  the  value  of  y  in  (2),  we  have 

4:^;  =  32;  +  3  +  I 

Uniting,  x  =  4 

2/  =  15 


And 

— ,  Ans. 

y     15 

12.     Let 

6x       A's  money, 

• 

And 

8?/  -  B's      " 

Then 

•  6x  -{-    y       I1200 

(>) 

And 

a;  +  8?/  —  $2550 

(2) 

Mult. 

(i)  by  8, 

48:?;  +  8?/        I9600 

Subtracting, 

47a;                  $7050 

X  =    I150 
6a;  =    I900,  A's  m. ;  )     . 
In  (i),  y  =  I300,  and     Sy  =  $2400,  B's  "      \ 

13.  Let  X  and  y  denote  the  numbers. 
Then  x  —  y  =z  14 
And                          X  -\-' y  —  48 

Adding  2a;  =62,    •*•  ^  =  3^ ;  [  ^^^^ 

Subtracting,  2?/  ==  34,    .*.  ?/=  17,  j 

14.  Let  it'  =  price  of  the  house, 

y  --     a        «       garden. 

Then  x  +  y  ^  $8500 

y  =  — 
12 

Substituting,  x  -\- --  —  8500 

Mult,  by  12,  etc.,  17a:  =  8500  x  12 

a;  =:  I6000,  house;  )    . 
And  ^  =  $2500,  garden,  \ 


TWO    unknow:n^    qua  duties.  91 


15- 


i6. 


Let 

40;       one  part, 

And 

6y        the  other. 

Then 

4^  +  6//  —  50                     (i) 

And 

3'^  +  5^-40                     (2) 

Mult.  (i)by3, 

122;  4-  18?/        150 

"      (2)  by  4, 

12a;  -f  2oy        160 

Subtracting, 

2y  -     10                   (3) 

Mult.  (3)  by  3, 

6^       30,  one;    )^^^ 
4ic       20,  other,  J 

Substituting  in 

(0, 

Let 

a?  —  A's  share, 

And 

y  —  B's     " 

Then 

X 

+  y  —  $1280                        (i) 

Anc. 

^x  —  gy                              (2) 

From  (2), 

7 

Substituting, 

9y 

7 

4-  ?/  —  $1280 

^W5. 


Clearing  of  frac.,  gy  +  iy  =  $1280  x  7 
Uniting  terms,  i6?/  =  $1280x7 

Dividing,  ?/  =  $560,  B's  share; 

Subst.  vahie  ?/  in  (i),      a;  =  $720,  A's     " 

17.  Let  X  =  age  of  elder. 
And  ?/  =    "     "  younger. 
Then  x—  y  z=  10                              (i) 
And  x—is  =  2{y—  15)                (2) 
Subt.  (i)  from  (2),  ?/— 15  =  2y  —  40 
Transposing,  ^  =  25  yrs.,  y'nger;  ]  ^^^^^ 
Subst.  25  for  ?/ in  (i),      ic  =  35    "     elder,     j 

18.  Let  X  =  value  of  ist  horse. 
And  y  =     "       "  2d      " 

Then     a:  4-  50  r=:  2?/  or     x  =  2y  —  50        (i) 

And  y  +  50  =:  ;r  —  15  or  x  :=z  ?/  4-  65 
Equating  the  values  of  x,  2y  —  50  =  y  +  65 
Transposing,  y  =z  $115,  value  2d ;  ) 

Subst.  value  y  in  (1),     x  =^  $iSo,      "      ist;  j 


92  SIMPLE     EQUATIONS. 

rage  119. 

19.  Let  X  =  distance  steamer  goes, 
And  y  =       ^'        sliip  " 

Then  20  +  ?/  =  a;  (i) 

Their  respective  distances  for  the  same  period  of  time  will 
be  m  the  ratio  of  8  to  7.     Hence,  forming  a  proportion, 

X     :     y     ::     S     :     y 

Reducing  to  an  equation,  yx  ^  8y 

Dividing  by  7,  '    x  =  —  (2) 

Substituting  in  (i),  20  -\-  y  =z  -^ 

Clearing  effractions,        140  -\-  jy  =:  Sy 

y  =  140  miles,  ship  ;       ) 

From  (2),  X  = =  160     ^'     steamer,  j 

20.  Let  6x  =z  greater. 
And                                          6y  r=  less. 

Then  3a:  +  2?/  =  13  (i) 

And  2X  —  $y  =    o  (2) 

From  (2),  2X  =  7,y 

Or,  X  =  M  (3) 


Substituting  in  (i),      — 


-i-  2?/  =  13 


Multiplying  by  2,  <^y  -^r  ^y  =  26 

y  =    2 
Substituting  ?/in(3),  a;=    3 


6x  =18,  greater;  )     . 
6y  =:  12,  less^         j 


TWO     UNKKOWN     QUANTITIES.  93 


21. 


Let 

^x  -^ 

part  hft, 

And 

6y  - 

"    carried 

away. 

Then 

X  -^  y  — 

28  feet. 

(I) 

And 

15a; 

-36^  - 

12    " 

(^) 

Mult. 

(0  by  I 

(2)  from 

5. 
■  (3),  ' 

15Z 

+  152/  - 

420    " 

(3) 

Subt. 

5^2/  - 

408  feet. 

• 
• 

y  - 

8    " 

And 

6y  — 

48    " 

(4) 

Subst. 

8  for  y 

in  (i), 

1 

X 

20    " 

And 

3^  - 

60    " 

(5) 

Adding  (4)  and  (5),     3a;  +  6?/  =  108  feet,  Ans. 

22.  Let  2x  =  the  lady's  age, 
And  y  =  number  of  verses. 
Then  y  z=  x  —  2                           ( i ) 
And  2X  +  y  =  43                                (2) 
Substituting,  2:^  +  0;— 2  =  43 

Uniting  terms,  $x  =  45 

X  —  15 

2x  =  3oyrs.,  her  age  ;)^^^^^ 
Subst.  15  for  a;  in  (i),  y  =  13  yerses,  f 

23.  Let  X  =  greater, 
And                                      y  —  less. 

Then  x  —  y  =  20  (i) 

X 

And  -  =    3  (2) 

y 

Fro-m  (2),  X  =  sy  (3) 

Substituting  in  (i),    3?/  —  y  ^=  20 

?/  =  10,  less;       I    . 
From  (3),  X  —  30,  greater,  j 


04  SIMPLE     EQUATIONS. 

24.     Let      X  =  No.  of  oxen,      and      y  =  No.  of  colts. 
Then  65:?;  +  25?/  =    720  (i) 

And  25a;  +  6sy  =  1440  (2) 

Mult,  (i)  by  13,       845:^  +  325?/  =  9360 
"     (2)  by  5,         125:2;  +  3252/ =  7200 


720ic  =  2160 

.-r  =    3  oxen  ; 
Subst.  in  (i)  and  transposing,  y  =  21  colts, 


Ans, 


25- 

Let 

X       digit  in  tens' 

place. 

And 

y         "      "  units 

'     « 

Then 

^  +  y  -\-  7  —  3^ 

(0 

And 

lore  -\-  y  —  18        loy  4-  i? 

(?) 

Reducing  (i), 

y  —  2X'-'j 

(3) 

"     (2), 

X      y  -  2 

Substituting, 

X  —  2iC  +   7            2 

X        5 

From  (3), 

y  -  z 

lo:?;  +  y        53,  A71S, 

26. 

Let      . 

6x       A's  entire  ca 

ipital, 

And 

Sy       B's      " 

(( 

Then 

6ir  +  5?/        I9800 

(I) 

And 

5z  _  4?/ 

(?) 

Or, 

X   ^y 
5 

' 

Substituting, 

^  +  5?/    19800 

Multiplying  by  5,  and  uniting  terms, 

49?/  =  ^9800  X  5 
y  =  $1000 
And  5?/ =  $5000,  B's;  I  ^^^^ 

Subst.  in  (i)  and  transp.,     6x  =  I4800,  A's,  ) 


TWO     UNKNOWN     QUANTITIES.  95 

27.     Let  X  =  part  of  purse  i  guinea  fills, 

And  y  =     "        "  I  c'ollar      " 


Then 
Anc. 

6x  +  19?/        I 

5^  +  4y  -  U 

0) 

Multiplying  (i)  by 

5. 

30X  -f  95y  -  5 

(2)  by 

6, 

302;  +  24^  —  If 

Subtracting, 

7i«/  —  5  - 
,/          I 

y      21 

34    _ 
■  2T   - 

_    71 
-    2T 

From  (2), 

5^  +  A  -  H 

Transposing, 

5-^    —    6  3   ■ 
X   —    ^j 

4 
~   2T 

A 

Now  if  I  guinea  fills  ^^3  of  the  purse,  it  is  plain  that  63 
guineas  will  fill  f  |,  or  the  entire  purse.  Reasoning  in  like 
nianner,  we  find  21  silver  dollars  will  likewise  fill  the  purse. 
Hence,     Ans.  $21,  or  6;^  guineas. 

28.     Let  X  =  greater, 

And  y  =  less. 

Then  .r  4-  ?/  =  « 

And  X  =  ny 

Substituting,  ny  -\-  y  =  a 

Factoring  and  dividing,  y  = ;  J 


Ans. 


And  X  = 


n+  i' 


96 


SIMPLE     EQUATION'S. 


THREE    OR    MORE    UNKNOWN    QUANTITIES. 


Page  121, 


2.     Giyen 


And 


Adding 
Mult,  (i 
Adding 
Mult.  (4 
"     (6 

Adding 


(2)  and  (3), 

)  ^y  2, 

(2)  and  (5), 

)  by  4, 
)  by  5» 

(7)  and  (8), 


5^  -  32/  +  22; 
2,x-\-  2y  —  4Z 

2X  -h  5«/ 
102;  —  6?/  +  42; 

13-^  — 4«/ 

8a;  +  2oy 

65a:  —  2oy 


28 

15 
24 


39 

56 

71 
156 

355 


73a; 


Substituting  7  for  x  in  (4), 

"  val.  X  and  y  in  (2), 


a;  =: 

y  = 


(0 

(3) 
(4) 

(5) 
(6) 
(7) 
(8) 


3.     Given 


And 


Adding  (2)  to  (3), 
Mult,  (i)  by  2, 
"     (2)  by  3, 

Adding  (5)  to  (6), 
Dividing  (4)  by  3, 

Subt.  (8)  from  (7), 
From  (8), 


a 


(2), 


2^  +  5^  —  3^  =    4 
AX  —  zy-\-2Z—    9 

5a:  +  6?/  —  2Z  ^n  18 


9^  +  32/ 
4a:  +  loy  —  62; 
12a;  —    ()y  -\-  dz 


—  27 
=     8 

=  27 


16a;  + 

3^  + 


.?/ 


35 
9 


13a; 


=  26 


a; 


(I) 

(2) 
(3) 

(4) 
(5) 
(6) 

(7) 
(8) 


=  ^  :  }•  ^?i5. 


THREE    OR    MORE    UNKXOWN    QUAXTITIES.  97 

4.  Given  2X  -{-  ^1/  —  4^  =  -^  (0 

X  —  ly  +  z^  —     6  (2) 

And  3a;  —  2?/  +  52;  =  26  (3) 

Adding  (i)  to  (2),         ;^x  -\-    y  —    z  =1  26  (4) 

Subt.  (4)  from  (3),  i^y  —  6z  =    o 

Or,  y—2z=:    o  (5) 

Bringing  down  (i),        2x  -\-  ^y  —  4z  =  20  (i) 

Mult.  (2)  by  2,  22:  —  4^  +  6;z  =  12 

Subtracting,  ^y  —  loz  =    8  (6) 

Mult.  (5)  by  7,  7y  —  14Z  =    o 

Subtracting,  42;  =    8 

2;  —  2 ;  j 
From  (5),  «/  =  4;  >-4/^s. 

"      (2),  x  =  S,  ) 

5.  Given  5a;  +  2?/  +  42  =  46  (i) 

32;  +  2^  +    2;  =  23  (2) 

And  10^  +  5^  +  4^  =  75  (3) 

Subt.  (i)  from  (3),        5^  +  3^  =  29  (4) 

Mult.  (2)  by  4,  i2:z;  +  8?/  4-  4^  =  92  (5) 

Subt.  (3)  from  (5),         2X  4-  sy ^J7  (6) 

«      (6)      "     (4),         3^ 

From  (6),  ^  =  3 ;  j-  ^47i5. 

"      (2), 

6.  Given  a;  4-    ?/+;;=    53  (i) 

x-^  2y  +  3^  =  105  (2) 

And  >^  +  3^  +  4^  ==  134  (3) 

Subt.  (i)  from  (2),  y  4-  2Z  =    52  (4) 

''      (2)     "      (3),  ^+    ^=     29  (5) 

"      (5)     "      (4), 
From  (5), 


98                                SIMPLE 

equatio:n^s. 

7.     Given 

zx  -h  42; 

57 

(i) 

2y         z  _ 

1 1 

(2) 

And 

5^  +  3i/  - 

65 

(3) 

>lulti plying  (2)  by 

4,             2>y  —  ^z 

44 

(4) 

Adding  (i)  to  (4), 

3^  +  8^  - 

lOI 

(5) 

Multiplying  (5)  by 

5,          15:?;  +  40?/ 

505 

(6) 

(3)  by 

3,          15a:  +    9^  - 

195 

Subtracting, 

z^y  — 

310 

^  - 

10;  > 

9;  h^ 

From  (2), 

z 

ins. 

"      (i). 

X 

7  J 

8.     Given 

X      y      z 

«  +  -  +  - 
234 

62 

(I) 

X       y      z 
-  +  -  +  - 
3       4       5 

47 

(4) 

And 

4'^5       6  - 

38 

(3) 

Clearing  of  frac. 

6a;  +    4?/  +    32  _ 

744 

(4) 

202;  +  15?/  +  122  _ 

2820 

(5) 

15a:  +   127/  +   10:2; 

2280 

(6) 

Mult.  (4)  by  4, 

2/\x  +  16?/  4-  125; 

2976 

(7) 

Subt.  (5)  from  (7), 

4-1'  +2/              - 

156 

(8) 

Mult.  (5)  by  5,        ] 

[Oo:r  +  75?/  +  60Z 

14100 

(9) 

"      (6)  by  6, 

^ox  +  72?/  -h  60Z 

13680 

Subtracting, 

io:r  +    3^              = 

420 

Mult.  (8)  by  3, 

12a;  +32/              — 

468 

Subtracting, 

2a;                       — 

48 

•  •           X  — 

24;  ) 

From  (8), 

y  - 

60;  V 

A7IS. 

"      (3), 

z  — 

120,   ) 

THREE    OR    MORE    UXKXOVrK^    Ql:A^■TITlES.  99 


12. 


Pofje  122. 

Giyen 

to  -\-  X  -\-   Z           10 

(i) 

X  -\-  y  -\-  z        12 

(2) 

tv  +  x^  y          9 

(3) 

And 

w  -\-  y  -\-  z        II 

(4) 

Adding, 

^w  -\-  Z^ -\-  Zy  -^  2,^        42 

(5) 

Dividing 

(5)  by 

3,       w  +  x  +  y  +  z        14 

(6) 

Subtracting  each  given  equation  from  (6),  we  have, 
w=2;     x  =z  T^;     y  =  4;     and     z  =z  ^,  Ans. 


3.     Given 

-  +  '        ^ 
X       y        6 

I        I         7 

2^       2;        12 

(I) 

(2) 

And 

II        3 

a;       2;       4 

(3) 

Adding, 

2       2       2        26 

a;       ?/       ^        12 

(4) 

Dividing 

/  \  1          III        13 

(4)  by  2,  -  +  -  +  -       -^ 
^  '    -^        X       y       z        12 

(5) 

Subtracting 

each  given  equation  from 

(5), 

we  have. 

16           14 

—    ._  •            —       • 

X        12 '        ?/        12 ' 

I 

z 

_   3  . 
~  12' 

Hence, 

X  -.  2\       y  —  z\ 

PROBLEMS. 

z  = 

=  4,  Ans, 

# 

Taqe  123. 

I.     Let 

X       age  of  ist, 
y  —    "    "  2d, 

And 

z  -    "     "  3d, 

100  SIMPLE     EQUATION'S. 


Then                               x  +  1/       27 

(0 

X  +  z       29 

(2) 

And                                y  +  ^       32 

(3) 

Adding,               tx  -\-  2y  +  2Z       88 

(4) 

Dividing  (4)  by  2,    x  +  y  +  z  —  44 

(5) 

y-    " 

And 

z  —     " 

Then 

.    7^  +  13^  —  I205 

I4:r  +     52;        I300 

And 

123/  +   20;2         I140 

Mult,  (i)  by  2, 

142:  -f-  26?/            410 

Subtracting  equations  (3),  (2),  and  (i)  from  (5),  we  have 
a;  =  12  yrs.;  ^  =  15  yrs. ;  and  z  =  ij  yrs.,  ^ws/ 

2.  Let  (c  r=  price  of  a  calf, 

"    sheep, 
"    lamb. 

(I) 

(2) 

(3) 

(4) 

Subt.  (2)  from  (4),     26?/—     52;  =     no  (5) 

Mult.  (5)  by  4,         1041/  —  202;  =    440  (6) 

Adding  (3)  to  (6),  116?/  =    580 

«/  =    $5,  pr.  of  sheep  ;  ) 
From  (5),  z=    $4,      ''    Vdmhs;)-  Ans, 

"      (i),  X  =  $20,      ''     calves,  ) 

3.  Let  a;  =  ist  number, 

«/  =  2d       " 
And  2  =  3d       ** 

Then  J»  +  2/  =  ^3  (0 

X  -\-  z  =z  16  (2) 

And  y  -\-  z  =  ig  (3) 

Adding,  2a;  +  2^/  +  22;  =  48 

Dividing  by  2,        x  -{-  y  +  z  =^  24.  .  (4) 

Subt.  (3)  from  (4),  a;  =    5,  istr;  \ 

"     (2)     «     (4),  ^  =    8,  2d;  V^^^5. 

"     (i)     "     (4),  2;=  II,  3d,    ) 


THREE    OK    MORE    UNKNOWN    QUAXTITIES.  101 

Let  6x  =  number  of  men  in  ist, 

yj  =         "  "  "       2d, 


(i) 

(3) 
(4) 
(5) 
(6) 
(7) 
(8) 


And 

2Z  = 

a           a 

"      3d. 

Then 

6iP  +  32/  +  22; 
3^  +  2/ 

=  1905 

2Z  —  60 

And 

ig  (i)  and 

(2), 

Z  -\-  2X 

—  sy    165 

Addii 

gx  +  4y 

1845 

Mult. 

(3)  by  2, 

4X  —  6y  -\-  2z 

—  —330 

Subt. 

(5)  from  1 

(0, 

2x-\-  gy 

=  2235 

Mult. 

(6)  by  9, 

18a;  +  8it/ 

_  20115 

ti 

(4)  by  2, 
acting  (8) 

from 

iSx  +    Sy 

=   3690 

Subtr 

'  (7),           73^ 

-  16425 

•••     y 

=     225 

From 

(6), 

X 

-     105 

« 

(2), 

z 

=     300 

6x 

—  630,  men  in 

ist;  1 

sy 

675,    "      " 

2d ;  ^  Ans. 

2Z 

—  600,    "      " 

3d,    ) 

5,     Let  X  =  price  of  ist, 

y  =     ''      "  2d, 
And  z  =     "      "  3d. 


Then 

122;+  izy  4-  142 

25 

(0 

10.^  +   17?/  +   112 

24 

(2) 

And 

6x  -\-  i2y  -\-    6z 

15 

(3) 

Mult.  (3)  by  2, 

12a;  +  24?/  +122 

30 

(4) 

Subt.  (i)  from  (4) 

iiy  —    2Z  — 

5 

(5) 

Mult.  (2)  by  6, 

60X  -\-  lozy  +  662 

144 

(6) 

''     (i)  by  5, 

60a;  +    651/  -1-  702 

125 

(7) 

Subt.  (7)  from  (6), 

3iy         4Z  - 

19 

(8) 

Mult.  (5)  by  2, 

22?/  —     4Z 

10 

Subtracting, 

^sy         - 

89.00 

\ 

V  —  80.60, 

2d;  ) 

From  (5), 

z  —  -^0.80, 

3d;[ 

Ans, 

"      (3)^ 

X       $0.50, 

i§t,  ) 

102  SIMPLE     EQUATIONS 


Let                        X       number  minutes  it  takes  A, 

y 

"     •  B, 

And                       0  -       " 

0. 

Now  if  A  can  fill  it  in  x  minutes, 

A  can  fill  -  of  it  in  i  minute, 

X 


B    " 

i< 

I 

~y 

a 

i(   ,         ii 

And         0    « 

a 

I 

z 

a 

(C     J             <{ 

Then 
And 

I          I            I 

X      y       'JO 

T           I              I 

X       z        S4 
I       I         I 

y      z        140 

(1) 
(3) 

Adding, 

2 

X 

2       2         28 
y       z        840 

2 
60 

Dividing  by  2, 

• 

I 
X 

III 

y       z        60 

(4) 

Subtracting  (3)  from  (4), 
Or, 


I        140  —  60 


X  8400 

I  _     I 
X  ~  105 
Clearing  of  fractions,  x=  io5min.,  A. 

Subt.  (2)  from  (4),       -  =  -,- ^-  =  , ^ 

^  ^  ^^        y       60       84        60  X  84 

A  I  24  _       I 

Or,  -  =  7 ^-  = 

y        00  X  84        210 

Clearing  of  fractions,  ,y  =  210  min.,  B. 

Subtracting  (i)  from  (4),  etc.,         z  =  420 'min,,  G. 

A  fills  it  in  105  minutes  ; 

B     "      "      210         "         y  A71S, 

0    "     '•     420       " 


THREE    OR    MORE    UNKNOWN     QUANTITIES.  103 

Denote  the  parts  by  to,  x,  y,  and  z. 

Then                        w  +  x  -\-  ij  -]-  z  —  %()o  (i) 

10  -{-  2   =z   2]!  (2) 

x—2^2y  (3) 

And                                                 ;  =  2^  (4) 

Multiplying  (4)  by  2,                     ^  =  4i/  (s) 

Adding  (5),  (3),  and  (2),    lo^x^z  =  Sy  (6) 
Subtracting  (6)  from  (i), 
Or, 


ii 


ii 


From  (2),  w  =  18,  ist;  f  ^^^^ 

"      (3),  •  :r  =  22,  2d;  r  ''*• 

"      (5),  2  =  40,  4th  J 

8.     Let  X  =  A's  distance, 

y=:B's 

And  z  =  C's 

Then  a;  +  //  +  2;  =  62  (i) 

a:  =  42+ 2;^      (2) 
And  20;  +  sy  =  i^^  (3) 

Subtracting  (2)  from  (i),     3^  +  52^  =  62  (4) 

From  (3),  2:r  +  3^  —  172:  =    o  (5) 

Mult.  (2)  by  2,  2X  —  \y  —    82:  =^  (6) 

Subt.  (6)  from  (5),  ly  —    9^  =    o  (7) 

Multiplying  (4)  by  7,         21?/  +  352:  =  62  x  7 

"  (7)  t)y  3,        2iy  —  27z=    o 

Subtracting,  622;  =  62  x  7 

.-.  =7  miles,  C's  distance;  j 

From  (7),  «/  =    9      "       B's        "  I  ^?^s. 

''      (2),  .T  =  46      '•       A's        "         ) 


104  GENERALIZATION^. 


Let 

4X  .  _ 

A's  n 

loney, 

2y  — 

B's 

a 

And 

3^  — 

C's 

a 

Then 

4x  +  y  — 

$100 

.    (•) 

21J  -\-  Z  — 

$100 

(2) 

And 

32;  +  a;  _ 

$100 

(3) 

Mult.  (3)  by  4, 

4X  -\-   12Z  — 

400 

(4) 

Subt.  (i)  from 

(4), 

I2Z  —  y 

300 

(S) 

Mult.  (5)  by  2, 

(2)7 

-  2y  4-  240 

600 

(6) 

Adding  (6)  and 

252;  _ 

700 

z 

$28 

And 

32  — 

$84. 

C's;) 

From  (4), 

4X 

$64, 

A's;[ 

Ans, 

"      (2), 

zy  — 

$72, 

B's,  ) 

GENERALIZATION. 
rages  124,  125. 

1.  a;  =  y  =  — ^  =  3  chickens,  Ans, 

b         IS 

480  .       . 

2.  X  =  — ^-  =  30  rods,  Ans, 

16 

576  . 

3.       X   =    -—     =:    12,    ^??5. 

48 

0!  61320  _  ^,  . 

4.  .^'  —  —  =r  ^i —  —  ciyL  years,  C  s  age,  ^W5. 

5.  a;  =     ~ — „  =  7  feet,  ^w*-. 
^  9x8' 

62730  , 

6.  X  z^ ^^    =  34,  Ans. 

41  X  45 


s  +  ^        392  +  18 

(/  =: = =   $)2  05 

2  8 


Ans, 


GENEKALIZATION".  105 


rages  126-127. 


o      ^        IS75  +  347        ^  .       .,         ,    \ 
o.    g  = =  I961,  As  part;  j 

/  Ans, 
I  =  157i_p_34Z  =  $614,  B's    " 

2150  4-  -546  ,       \ 

9'    9  =  — ^- -^-^^^  =  1248  votes;  j 

.  )  Ans. 

,        2150  —  346 

^  / 

«^  8  X  12       96 

10.  X  =  — -—7  =  — — — -  =  —  =  44  days,  Ans, 

a  -\-  b       8  -I-  12        20       ^      -^  ' 

9  X  15        135  .  , 

11.  X  =  ^— — ^  =  -^  =  5I  hours,  A?is. 

9  +  15         24        ^^ 


40  X  50         2000  .  ,  . 

12.  .T  = — ^^  = =  2  2#  hours,  Afis, 

40  4-50  90 

13.  ;j  =  br  =  748  X  .09  =  $67.32,  ^?i5. 

14.  p  =  45385  X  .20  =  9077,   ^;?s. 

15.  ^^  =  2763  X  .375  =  11036.12^,  Ans. 

16.  p  =z  1587  X  .37  =  587.19  bushels,  Ans. 

b        2700 


17.    /•  =  4r  =  i^r  =  -^^f,  ^«6'. 


2C5.2 

18.  r  =  ^-  =  .40,  .l7i5. 

03 

291 

19.  r  =  -|-  =  .60,  A71S. 

485 

r        .25        *^ 


106 


INVOLUTION. 


21.       h  = 


37- 


TageH  130-133, 

-^  i=r   I37500,    J;Z5. 


22. 

^ 

23- 

<-? 

<T^ 

24. 

a 

25- 

a 

26. 

i 

27. 

i 

28. 

• 

I 

29. 

a 

30- 

a 

31- 

a 

2500 


=  $31250,  ^l?iS. 


^4^i5. 


32.   p  = 


33'    li  = 


h  (i  -\-  r)  =  2500  X  1.08  =  I2700,  B's ; 
b{i  — r)  z=  2500  X   .92  =  $2300,  O's, 
^  (i  +  r)  =z  4500  X  1.25  =  I5625,  A71S. 
b  {i  — r)  =  2750  X   .67  =  1 842 1  acres,  A^is. 
prt  =  465  X  .06  X  2  =  I55.80,  Ans. 
1586  X  .08  X  ij  =  1 1 90.3 2,  Ans. 
3580  X  .07  X  5  =  'ti253,  Ans. 
p  +  prt  =  364  +  364  X  .15  =  1^418.60,  A71S. 
4375  +  4375  X  .20  =  $5250,  A71S. 
2863.60  +  2863.60  X  .35  =  13865.86,  Afis. 
a  1500 


I  -\-  rt        1. 1 2 


—  $i339-29,  Ans, 


t 

t 


=   $222,222,    Ans, 

1.35 

O'  —  V  525  525  T  ^ 

=  --^ — -  =  --^  =  2I  yrs.,  Ans, 

pr  3500  X  .06        210 


«7j  =  II  X  3  =  yy  =  3  :  i6/j  o'clock,  p.  m.,  Ans, 
H  X  6  =  fl  =  6:32^  o'clock,  P.M.,  A71S. 


38.     t  = 


if  X  9 


108 

II' 


9 : 49 j^j  o'clock,  p.m.,  A71S. 


{ahcf  —  aWc^. 
( — abcY  ::^  aW(?. 

(cibcY  —  a^b^c\ 


INVOLUTION. 
Page  137. 

7.  {6aWY=  2i6«W 

8.  \sfiWcY  =  62sa^'^b^c\ 
.  9.   (2«2^>f2)6  ^  64^12^,^2. 

10.   {abcdY  =  a^b^cfd^. 


lis  VOLUTIOX. 


107 


12 
15 


Page  13S. 

The  5tli  power  of  {a  +  hf  =  {a  +  ly^. 
The  2(1  power  of  (^^  +  Z*)"  =  {a  +  Z-)"-". 
The  ?/th  power  of  {x  —  y)"'  —  (x  —  i/)'"". 
The  nth  power  of  (:<;  +  ?/)-  =  (:t'  +  ?/)-". 

16.  The  2d  power  of  (a^  +  Z-^)  —  (^^3  ^  ^3^2. 

17.  The  3(1  power  of  (a^Vi^)  =  aWi^. 


^aI)'^Y__  27^3^6 


\  2a  I 

/2a^cy_ 


8«3 
\6a^¥c^ 


hjtl^V-_  49«^ 


^'   \  xy'' '  ~  x^'y"'"'  ' 


or 


x"y'^ 


26.   (x  -\-  2y  -{-  2)3  z=x^  -\-  6x^y  +6.i;2+  i2a;«/^+  24;/-?/+  12:?: 
+  8z/3  +  24?/'^'  +  24?/  +  8. 


Prt</e  142. 

1.  («  4-  Z')4  =  «4  ^  4^^3^  ^    5^2^2  4_   4^^J3  4.  J4. 

2.  (7  —  hf  =  ft5  _   5^4^  4.    10^3^2  _  io^^2^3  _^   ^^^4  _  J5, 

3.  (r  +  fZ)7  —  c7+  7C6C?+   216-5^2  ^  35^4(^3 _^32^^/4_|.  21^2^/5 

4.  (a;  +  ?/)6 = a;6  +  6a:57/  +  1 5 x^y^  +  2 o^'^,?/^  +  1 5 .7-2^4  _|_  ^^y5  _|_  ^^ 

5.  {x  —  yy  =  x^  —  'jx^y  +  2 I2;y — 352Y  +  35-^^— 2 1:^'^ 

6.  {y  +  zy^  =  ?/io  +  10^9^  +  4St/z^  4-  120/^3  +  210/2* 

4-   252/25  _|_   210/26  _p    I20?/V  +   45/2^  +    10^2^  +  2^0. 

ir 

7.  (a  — by  =  a^—  ()a%+T^6cvl?  —  ^4a^¥-\-i2()a^¥—i26a^h^ 

4-  84rt3Z,6  _  36«2^T  ^  c|rtZ/8  —  h\ 

'  8.   (y;?  +  ny^  —  m^^  +  iim^^n-^- ^^m^}i^  +  16^111^}}^ -\- 2,7,0)11^11^ 

+  462111^)1^  -\-  462111^11^  +  330???'^;^'  +  I  65  y»3;^8  _|_  55/7|2^^9  _^   J  J  „^^^io 

4-  n^\ 


108 


INVOLUTION. 


9.  (^x  —  t/)^  =  x^^  —  i2X^^y  +  66.^iy  —  22o:fy  +  495:^^ 

—  7922;'^^  -f  9242-^^6  —  i<)2X^y'  +  495rcy— 2  2o;ry +  66a;y^ 

—  i2xy^^  +  y^. 

■yz  ^—  I 

10.  (a  4-  b)"  =  a"  +  ^^a"-^^  +  n a^'-^i^ 

2 

2  3 

4-  n X  X a''-^  h\  etc. 

234 

13.   {x  +  1)3  =  .t3  +  3.i-2  4-  3;7-  +  I. 

14.     {})  _   i)4  =li  —  4^,3  ^  6^2  _  4J  +    I. 

15.   (i  —  a)5  z=  I  —  5^  -f  106?'^  —  loa^  +  5^^  —  a^. 


16.   (i  +  a)"  =  I  +  7^r/  +  n 


n—  I 


c^  +  ?z 


n  —  I     n  ~  2   . 

■ X a^ 


+  etc. 


Pftge  143, 

17.  (x-^y  +  zy  =  x^-\-  ^x^y  +  ^xh  +  3.t?/2  +  6xyz  -\-sxz^-{-  y^ 

20.   (2;  +  ^  +  2;)"  =  a;2  +  «/2_j_2;2_^2:?;?/-r22;2;4-2^^,  or 

=  2^2 _|_  22!  (?y  _(-  2;)  _|-  7/2 _|_  27/2;  +  2^,  J ns. 

21.     («  — ^4-c)2  =r  a2  +  Z>2  +  c2— 2a^+2r^6'— 2Z><?,   01* 

=  a^—2a{b  —  c)  +  i^—2bc  +  c%  Ans. 

22.   (<i  +  a:  +  ^  +  2;)2 

=  «2  _|_  ^2  _|_  ^2  _|_  2;2  ^  2fl!a:r+  2fl5?/  +  2az  +  2.1'?/  +  2XZ  +  2?/^,  or 
=  a^  +  2a{x  +  y  +  z)-\-x^+  2x(y  +  z)+y^+  2yz  -f  z^,  A  ns. 


-  ("-y={^r= 


Page  144, 

T,a  4-   2\2         9^/2  +    12(7  +  4 


^^-  (-3  =  (^) 


3      ^  9 

2      4<72  —  ^ac  +  f^ 


' ,  Ans. 


,  Ans. 


IN^y  OLL  TION. 


109 


,6.     (-  ^;  +  .au)'  =  ^J^^^ 


27 


•     (-^  +  3'^) 


;^6  —  i6SaI)C  +  igOa^h^     . 
.  j±ns» 

49 


m 


b^  —  6'bmxy  -f-  gnAr^if      . 


MULTIPLICATION    AND    DIVISION    OF    POWERS. 


3 

4 

7 
8 

9 
10 


Q!^  X  n^^  =  «^. 


Page  14:5. 

5.     r"^  X  l^ 


X 


"5    v/      !• — 3  /y — 8 


a~^ccl  X  «V^^  =  a^c^(K 
a^y~h^  X  a~^y~^'^  =  a^y'h"^. 


=  b-^ 


6.     «"^  X  rt"  =  «'"+". 


12. 

13- 
16. 

17. 

18. 

19. 
24. 

25- 


rage  146. 


'7'~S     *     '>»3  —  Q*    11 


14.     ¥-^W  =  t^. 


15- 


i2a%~'^c  -r-  ^d^b~h^  =z  4abc~K 


.'-5 


c  ^  =.  c  ^. 


ax 


a 


=  -^ ,  Ans. 

y         xhj 


/-4 


a    _  ay        . 

~,  —  — ^ —  1    A  )IS' 
4  h 


by 


ad~^        a         . 

26.      — ^-  =:  -,^-7;  ,    ^^^5. 


iC^ 


rZ5.r2 


27, 


—^— ,  Ans. 

ax""         a 


no 


EVOLUTION". 


EVOLUTION. 


Pages  14S,  140. 


13.  as. 
5 


14.  x^. 

15-  y'- 

16.  ai  =  a-^,  Ans. 

17.  a-  =  a'^,  Ans. 

18.  r/s  r=  a'^,  Ans. 

19.     Js  —  ^-8^    A7IS. 

20.  2;^  =  2:1'^,  A71S. 

21.    «/T  ^  Z/*"^,    ^^5. 

3.  \/r/  =  a^,  Ans. 

4.  vV  or  a  =  «T^  yi?^5. 

5.  '\/4xy  =  4a:r3?/3,  ^7is. 

6.  '\/Sa^O^  =  2asb%  Ans. 

7.  \^ 2'jabc  ^  TfCt-^b^c^,  Ajis. 


8.   'v/i6a4'"  = 


9.    \^yc\v^^ 


2a'"\  Ans. 

:^sa'!>x^,  Ans. 
10.   Vs^ci^b'^  =  6^2^,  ^«5. 

2^x-^y^,  Ans. 

:  S(fib%  Ans. 

(13)^x^1/^,  Ans. 
=  'jx^y^  Ans. 
==  T^a^b^,  A71S. 


11.  V  2.1'^^'^  = 

12.  \/64«!^<^®  = 

13.  \/isxy  =z 

14.  V49^y 

15.  ^ 2^aW 


16. 


49^^ 


7a;' 


=  -^ ,  Ans, 


64/      8i/ 


Page  152 1 

X  -{-  2. 

a  —  I. 

I  +  X. 

X  +  ^ 


a 


3- 

I 
2* 

b 

X  +  -' 
2 


Page  153. 

x^^2xy-[-y^-\-2xz-\-2yz  +  z^  (  x-\-y-\-Zi  Ans. 

x? 


2X-\-y  )  2xyAr\f 


2X+2y-^z)  2XZ  -\-  2yz  +  z^ 


10. 


a' 


^■^4ab-\-4b^-\-2a—4b+i  («  — 2^+1,  A?is. 


2a— 2b  )  — 4ab  +  4b^ 


2a — 4^4- 1  )  2a — 4^+1 


EVOLUTION 


111 


II. 


a*  +  4U^b-\-4^—4a^—Sh-{-4  (  ^^4-2^  — 2,  Ans. 


a" 


2a^-\-2b  )  4C(?b-\-4lP' 

2(1'^ -\- 4b  — 2  )  — 4d- — 8Z'H-4 


12. 


I  —  4^'  +  4¥  +  2x—4l>^x -\-x-  (  i—2U'-\- Xy  Ans. 


13- 


2-2^/2)   _4/,2  +  4^4 


2 — 4l^-\-X  )  2X  —  4Wx-\-x^ 

4a^— 161^-^2401^— i6a-T 4  (  2^^ — 4^f  +  2,  ^W5. 
4«4 


4^2 _4f^  )  —i6a^-\-24a^ 
4«2 — 8«  +  2  )  8rt^ 

8^2. 


i6rt  +  4 
i6rt  +  4 


14. 


a 


a^ 


2_ 


4 


2 


« ,  Ans. 


2a 


-n- 


7    .    ^' 
4 

—  «J  H 

4 


15. 


2rc 
IT 


X^ 


2+^ 

X^ 


9       I 


-,  Ans, 

y     ^ 


f 

y 

X 

2   4-   — 


Note.'       — 2-= =  —  2  x  —  =  — -,  second  term  of  root ; 

y  2.1*  ^ 

Also,     —  X— ^=— 2;    and     — "^x--^  — • 
y  X  X  X       x^ 


112 


UEUUCTION     OF     RADICALS. 


REDUCTION     OF     RADICALS. 


Case  I,  Page  156. 

6.  aVlf' 

7.  2aV2b. 


8 

9 
10 

II 

12 

13 
14 

15 

16 

17 


6^/36  X  7«^^ 

3r^V  2C. 

2Ifl^\/l   —3^. 

3«V^^- 
6«\/i3^- 
Vi584a^=:A/i44«^x  11 


2 

3 

4 

5 
6 

7 

8 

9 

10 

II 
12 

13 


3/  ft^Z'^c^ 
27 


V  27  («  —  h)\ 


V{a  —  If. 


Va""\ 


2. 


Case  III,  Page  158, 

a^  ^  a^  =^  (a^)^ ; 


Ans. 


3I  _  34  = 
3 


=  (125)^,       ) 
^2  =i  a^  =1  {a^^;      \    . 

»t   ==   66    =:    (t296)«,  ' 


ns. 


V5  =  V'S^  =  V^i5625; 
^2  =  A^73  =  v^8, 


ADDITION     OF     RADICALS, 


113 


6.  'v^^    and     1/1252;^. 

7.  \^64a^     and     \/4.a\ 

8.  V^«    and    V^. 

9.  (^")2"''    and    (^2)2"". 


10.  \/{a  +  ^)2    and     Via  —  b)K 

11.  v^(ic  —  yf    and    v^(.c  +  ?/p. 


3- 

4- 


Case  JF,  Pagre  159, 

(3^)^    and    (42)^ 
(aio)^    and    {b^^)k 


^  4 

3-^^  =  S  X  ^  =  4; 

4  ^ 


I       3  _  I       4 4 

3   '    4~  3       3~  9 


Hence,  (a^)^    and    (S^)^,    ^?is. 

6.  (a^)^    and    (^tI)!, 

7.  (««)"     and    (^'^/'. 


6. 


7- 


ADDITION     OF     RADICALS. 


Page 

160. 

'v/12         2^3 

5. 

'\/20    =    2^/5 

A/27  _  3\/3 

V48            4V3 

5\/3,  ^ns. 

2A/5+4V3.  ^^^5- 

2^/W  - 

2bVb 

i^M  = 

3a  Vb 

{2b  +  3a)  a/^,  ^W5. 


aVsct^b  = 

cVzjnb  = 


a^Vsab 
3C\/3ab 


{a^  +  3^)  V3«f^.  -^^^5- 


114 


SUBTRACTION     OF     RADICALS, 


8. 


lO. 


II. 


12. 


13. 


gxy  2a 


(9a;  +  Za)  \^  za,  Ans. 
3V^54  =     9V^2 

4'V^I28   :=    161^2 

25  ^^2,  Ans, 


7^/243  =  7V81  X  3    =    63A/3 
5V363  =  5V121  X  3  =    55V3 


1 18  A/3,  Jws. 


a 


\/Sib  =    gaVb 
^aV49b  =  2ia\/i 


^oa\/b,  A71S* 

sV^  = s^^^y 

4x^vy  +  5^a/^'^j  ^^^s. 


SUBTRACTION     OF     RADICALS. 


2. 


Page 

161, 

4\/ii2  = 

=  16^/7 

1/448  = 

.     8a/7 

8^7,  J7^ 

V480  _ 

W30 

4A/63 

I2a/7 

'41/30  —  I2V7>    ^^?^« 


MULTIPLICATION     OF     KADICALS. 


115 


4V320  =  32^/5 

—  5V30  =   —  20^/5 


52^/5?   ^^^5- 
2IiC  a/^ 


(212:  —  10)  Vctx,  Ans, 


6.     s\^a  +  b  —  3V^fl^  +  <^  =  2VCI  +  ^,  ^^^s* 

8.  3'V^25oJ^^  =  i^b'\/2hx 

2'V^54^%  =    ^hV~2bx 

<^l)'^2lx,  Ans, 

10.  5^1  =  4V3  =  ¥V3 

2^/1  =   IV3    =     f  V3 

i|a/3j  -4ws. 


MULTIPLICATION     OF     RADICALS. 

rage  162. 


4.     5^/18  =  15^2 


II. 


abx. 


< 

3'\/2o         6v5 

6.      V«^  —  b^' 

9o\/io,  v4;z6'. 

7.     ^/acxy. 

s. 

«i  —  V«^;     Va^  X  Vc  —  «V«f,  Ans. 

BO. 

1                 J!L 

aj»       it'"'" 

7^"' 

mil  /  J 

7V^4  X  3\^4  =  2i\^T6  =  42^/2,   J/?,9. 


J2.       130!, 


116 


INVOLUTION     OF     KADICALS. 


13. 
14. 

16. 

17. 

18. 


19. 


6. 

4ax. 

2a/|  X  2VI  =  4VA  =  V5,  ^^«. 

Wi  X  sVi  =  i2'\/3\  =  2V5,  A71S. 

00  4. 

(m  +  ^^)^  X  (m  +  ny^  =  (in  +  nY', 
(m  +  w)3  =  (pi  4-  7?)  V m  +  n,  Ans, 
/gad  2db  Af/ 


DIVISION     OF     RADICALS. 


Page  164, 

or,  =  aVs(i'  (Art.  306.) 

5.  sVb^' 

6.  (r«2  _^  a;)i 

7.  i2(fl5?/)i 

8.  3&\/i?J. 


i5« 


Wl, 


10.        2« 


v^. 


2            I 
II. 

n      n 


n 


(a  +  b)^,  Ans. 

12.  3^/25:^=  153;^/^,  ^^?*% 

13.  ^/x  —  y,     (Art.  128.) 


14.  8 a/8  =  8^/4  X  2  =  16^/2,  ^/i^. 

15.  2^256  =  2  X  16  =  32,  ^WA-. 


INVOLUTION    OF    RADICALS 

Page  164, 

3.  «i 

4.  (3A/2a:)2  _  ^  ><  2a;  =  18:?;,  ^?J5. 

5.  Sa. 

6.  (    a/2:c)   —    -  X  2.r  a/2^  =  -  V^,  A?is. 

\2  /  b  4 

4y  - — 1    =  (2xVcty  ==  8a^3y^^  ^^^^^ 


EVOLUTIOK     OF     HADICALS  117 


8.     [sV     I    = =  9^^  -^^is. 


y  ~    9 
9.    CI?  +  2(1  Vy  +  y,  (Art.  266). 


2 
3 

4 

5 
6 

7 

8 

9 

10 
II 

12 


EVOLUTION    OF    RADICALS. 
Pafje  165. 

2,^/a^  =  3^«,  Ans. 

isV^y)^  =  {Vgxy)^  =  "s/^xy,  Ans. 
(2hV^)^  =  (a/8^)^  =  \/Sb%  Ans. 

(128^^)^  =  yz'^a^p  =  20^^,  Afis. 

"Va^b^  =  cM^,  Ans. 

(4«2y^)i^  =  (V32«^)^"  =  ^/ 120.^  =  \/2a,  Ans, 
i  i  i 


REDUCING    A   RADICAL   TO   A   RATIONAL 

QUANTITY. 

Case  I,  Page  166, 

3211 

4. =  -;  a%  Ans, 

3       3        3 

5-      ^3^3  X  a^c^  =  ac;    hence  ^^^c^^  Ans, 

6.     («  +  h)^  X  (rt  +  b)^  =  (a  +  b);  hence  (aj  4-^)5,^^5. 


118  REDUCTIOK     OF     RADICALS. 

7.     ^/  aWc  —  ah^/ac 

itb\Uic  X  ^/ (ic  —  c^hc.     Hence  ^ac,  Ans. 


8.  v(i«J  +  2/)^  X  \x-\-i)  =z'x-\-y.    Hence  Vx-\-y,Ans, 

9.  \^{a  +  Z?)2  X  Vci-^b  =  a  +  b.     Hence  \^a~i-  d,  Ans. 


10.     Va  -{-  b  -\-  c  X  Va  ^b  +  c  =  a -{-!)-}- c.      Hence 
Va  +  b  -\-  c,  Ans. 


3 

4 

5 
6 

7 
8 

9 

10 


Case  JJ,  Paae  167. 

X  —  4^9. 

{V9  —  V6)  (V9  -h  V6)  =:  9  —  6  =  3,  ^?^5. 

V;  —  Vci. 

(6  —  Vs)  (6  +  Vs)  =  3^  —  5  =  3h  -4.ns, 

Vs^^  +  Vs^' 

(Va  —  Vs)  {Va  +  Vs)  =  «  —  hi  ^^«- 

3A/^  —  a/8. 

4^2^  +  5  a/^. 

Case  III,  raf/e  168, 


^/x       yx^  ^ 


3/-  ^  ~37^^  —  — IT"  J  -4?^5. 


yc      ,yc^  ^ 

,       a/o;  +  a/v       a/^  4-  A^V       X  ■\-  2  Vxy  ±  1/       . 
6.  — /  X  — = p  =  — ^ ^i-^ ,  ^^5. 

A/ic  —  Vy       yx  4-  A^^  ^  —  y 

X  ^/a  +  y^o       xiy^a  4-  a/c)      . 

X  -— = —  =  — ^ -,  Ans, 


a/«  —  Vc       Va  +  A^c  ^ 


RADICAL     EQUATIOXS.  llO 


8  I         ,     I  —  Vs  _  I  —  A/3  _  I  —  A/3 

I  +  a/s       I  —  A/3         ^_~  3  —  2 

V3  —  I      ^ 
=r  — ,  Ans. 

2 

Note. — Changing  the  signs  of  both  terms  of  the  fraction  does  not 
alter  its  value.     (Art.  166.) 

_V3__  ^  3  +  Vs  _  3V3  +  3  _  3  (A/3  +  0 

3  —  V3        3  +  A/3  9  ~-  3  6 

A/3+1       . 

= ,  A?is. 

2 


RADICAL     EQUATIONS. 
Par/e  KiO. 

4.     Given  a  -\-  \/x  -\-  c  ^  d 

Transj)osing,  V  '^'  =  d  —  a  —  c 

Involving,  ic  =  (fZ  —  a  —  cY,  Ans. 


5- 

Given 
Involving, 

\^x  -j-  2        3 

X  +  2  —  27 

Given 

Transposing, 
Dividing, 
Involving, 

X        25,  An 

6. 

3VX  —  4  +  5-71 

3VX  —  4        2\        1 

A/a;  — 4       1 

^       4-11 

7. 

Given 

V4  "■' 

Dividing, 

Vh  » 

Involving, 

5  =  '. 

X       256,  ^^s. 

120  EADICAL     EQUATIONS. 


8. 

Given 

^/2X  +  3  —  6  —  13 

Involving, 
Transposing, 

2^"  +  3  —  6        2197 

2X           2200 

X        1 100,  Ans, 

9- 

Given 

V(^  —  4        3 

Involving, 
Given 

X  —  4  —  27 

X       31,  Ans, 

lO. 

2\/^  —  5    _   4 

Dividing, 
Involving, 

Vx  —  5        2 
X  —  s  —  1^ 

X       21,  Ans, 

II. 

Given 

5\/7     30 

Involving,  etc. 

2; 

--30 

X  _  252,  ^WS. 

Given 

rage  170. 

12. 

i-\-V2ax-}-b        „ 

Multiplying,  etc.,  \/2ax-\-b  =z  b^—a 

Involving,  2ax-\-b  =  b^—2ab^  +  a^ 

Transposing,  2ax  =  b^—2a¥-\-a^—b 

b^—2ab^-\-a^ — b    . 

X  ■= ,Ans. 

2a 

n-  y  —  (^y     "^y 

14.  Given  ^ — — -  =  — -' 

\/y  y 

Clearing,  etc.,     y  (y  —  ay)  =  y 

Dividing  by  ?/,         y  —  ay  ■=.  \ 

Factoring,  «/  (i  —  «)  =  i 

Dividing,  y  = ,  Ans, 


RADICAL     EQUATIONS.  121 


15.     Given  x  +  V«-  +  x-  — 


20.^ 


Clearing  of  frac,  x\/ci^  +  x'-\-a^-]-x^  —  2^2 


Transposing, 

x^/c^  +  a:;2       ^2  _  ^^ 

Involving, 

a^x'^ -\- x^  —  a^     2a^x^  +  x^ 

Uniting, 

^a^x^       a^ 

Dividing  by  a^, 

y 

^2  -     ''' 

Extracting  root. 

• 

3 
X       aVi,  A71S. 

17.     Given 

c  +  12        2  -\-  -y/x 

Involving, 

ic  +  12         4  -f  ^^X  +  X 

Transposing, 

4\/x        8 

Dividing, 

Vx        2 

Squaring, 

VI 

X       4,  Ans. 

18.     Given 

X  a/^  +  2         2  +  V5X 

Then 

Vs^  +  10  _  2  +  Vs^^' 

Squaring, 

Sx  +  10  —  4  +  Ws^  +  S-'^ 

Uniting, 

4V5-^'  ■     6 

Dividing, 

Vs^      2 

Involving, 

.     5^       1 

19.     Given 


*C/    — -■■      2  0?      "^^  /^t) • 

^/x       X  —  ax 


^  Vx 

Clearing  of  fractions,  x  =  x(x  —  ax) 

Dividing  hy  x,  (i  —  a)  x  =  i 

—       ^         J 

•  •  X  —  «  jfxii/Ot 

I  —  a 


133  PUKE     QUADRATICS. 


PURE     QUADRATICS. 
Page  17  S. 

2.  Given  32;^  —  5  =  70 
Transposing,  etc.,  rc^  ==  25 
Extracting  root,  ^  =  ±  5. 

3.  Given  9^2  _j_  g  _  ^^^  -}-  62 
Transposing,  etc.,  a:^  =  9 

X  =z  ±i  3,  Ans. 

4.  Given  53^2  +  9  =  2x^  +  57 
Transposing,  etc.,  a;^  =  16 

5.  Given  6^2  +  5  =  4^2  ^  55 
Transposing,  etc.,  x^  =^  25 

X  =:    ±^  5,    ^^S. 

6.  Given  ^ h  35  =  3^'^  +  7 

4 

Mnltiplying  by  4,       53^2  -f-  140  =  120;^  +  28 
Transposing,  etc.,  .^^  _  15 

X  :=  ±i  4,  Ans. 

^ .  2a;2  +  8        a:^  —  6 

7.  Given  = 1-  1; 

'  10  10         ^ 

Multiplying  by  10,         2x^  -j-  S  ^  x^  —  6  -\-  ^o 

Transposing,  etc.,  x^  =  ^6 

X  =z  ±  6,  Ans, 


8.     Given 


X  X       4 

42       a; 
Multiplying  by  4X,  x^  =  2.7^  —  16 

Transposing,  rc^  =  16 

a;  =  ±  4,  ^7ic^. 


t>tJKE     QUADRATICS.  123 


9- 

Given 

X       2        X       s 
2       X        3       X 

Multiplying  by  6x, 

3.^2  4-  1 2  —  2:c^  +  1 8 

Transposing, 

x^  -  6 
X        ±  V6,  A71S 

lO. 

Given 

2X^  +    12            SX^—  37 

Transposing, 

x^        49 
X        ±7,  A71S. 

II. 

Given 

7^^  —  7  —  3^^  +  9 

Transposing,  etc.. 

x^       4 

X        it  2,  Ans, 

12. 

Given 

«V  -  «4 

Dividing  by  a% 

X        ±  «,  ^ws. 

13- 

Given 

(ic  -1-  2)2       4a;  +  5 

Or, 

ic^  +  4X  -\-  4        4^  +  5 

Transposing, 

X^  =    I 

a;        ±1,  Ans, 

14. 

Given 

6x^  —  12 

iC''  —  I 

4 

Multiplying  by  4, 

4x^  —  4        6x^—12 

Transposing,  etc., 

x^       4 

.T       ±2.  Ans. 

15. 

Given 

a- (2^ +  9)       30^  +  6 
30                  10 

Multiplying  by  30, 

2X^  +  9^'  —  90^+18 

Transposing,  etc.. 

ic2       9 

iC  _    ±  3,   ^725. 

16. 

Given                  — 

4- 

5       ,       5      _8 

—  X       4  +  X       3 

Clearing  of  fractions. 

60  +  l^X 

+  60       15.'?:  —  128       82^2 

Transposing,  etc.. 

x^  -  I 

X  —  ±  ij  Ans. 

124  PtJR'E     QUADEATICS. 


Page  17  S — Continued. 


^.                              ax^  (a  —  2) 
17.     Given  ^, =  I  ^x 

'  I  -\-  X 

Clearing  of  frac,      a^x^  —  2ax^  =  i  —  x^ 
Transposing,    a^x'^  —  2ax^  -}-  x^  =z  i 
Factoring,         (a^ —  2«  +  i)rr2  =  i 

Dividing,  x^  =  -z 

°  «-*  —  2«  +  I 

I 

«  —  l' 


— r       4^ 

Dividing  by  2,  and  squaring. 


19.     Given  2a/^^  —  5  = 

^  3 


,2       -        4^' 


i^  —  5  = 

9 

Multiplying  by  9,         92:^  —  45  =  4^^ 
Transposing,  etc.,  a:^  =  9 

20.  Given  2^/-'^^  —  4  =  4V<'«^ —  i 
Dividing  by  2,  and  sq.,    x^  —  4  =z  40,^  —  4  '• 

Transposing,  x^  =  40^ 

X  =  ±:  la,  Ans, 

21.  Given  V^  -{-  c  = 

Mult,  by  denominator,    x^—  c^  =  d^ 
Transposing,  x^  ^=  c^  -\-  d^ 

X  =  ±,'\/ d^ -\- (P,  Ans, 

22.  Given  A  / =  vx 

Squaring,  etc.,  ^x^  —  i  ^  x^ 

Transposing,  etc.,  x"^  :=  \ 

•  •  X    —     ^t    2  >    -^  '^i5» 


PUKE     QUADRATICS. 


125 


23.     Given 


Squaring,  etc., 
Transposing, 


24.     Given 


Vic 


—  z=  \/x  +  a 


a 
h^  = 

x^  = 


x^  —  a^ 

«2  +  0^ 


X 


=  ±Va^  +  P,  A 


ins. 


24 


Va:  +  10 

Clearing  of  fractions,  24 

Squaring,  576 

Transposing,  x^ 


=:::    'S/ X  —   I O 


V^^  — 100 

:   J?'  —  100 
:   676 

X  =  ±  26,  A71S, 


PROBLEMS 
Page  174, 


1.  Let 

Then 

Clearing  of  fractions, 

2,  Let 
Then 
Multiplying  by  4, 


X  =:  the  number. 


XX 

-  X  -  =  108 
3       4 

x^  =  1296 
X  =  -t  36,  Ans. 

X  zzz  the  number. 

25-—  =  9 
4 

a;2  ^  64 


Let 
Then 


ic  =:  ±  8,  Ans. 

X  =  No.  of  rods  on  one  side. 
a;2  =3  1600  sq.  rds.,  area.     (Ax.  10.) 
X  =  40  rods,  Ans. 


Note. — It  is  advisable  for  tlie  pupil  to  represent  the  area  by  a 
diagram,  in  this  and  like  problems. 


Let 
Then 


X  =  length  of  side  of  square. 
x^  =  50x18  =  900  rods,  area. 
X  —  ^o  rods,  Ans, 


126 


PURE     QUADRATICS. 


8. 


Let 

2X       one, 

Then 

Sx       the  other. 

By  conditions. 

] 

[ox^       360 

Dividing  by  lo, 

x^       ^6 

• 
•  • 

X        ±6 

And 

2X           12;)       . 

]■  Ans. 
5^  —  30.  1 

Let 

X       No.  of  dollars. 

Then 

a^- 

-7  —  29 

Transposing, 

x^  —  36 

• 
•  • 

X       16,  Ans. 

Let 

X       number. 

Then 

X 

8 

X 

X  -  X 

5 

I 
16 

Clearing  of  fractions, 

x^       6400 

• 
•  • 

X       80,  Ans. 

Let 

x       less, 

Then 

4X       greater. 

And 

4X^       900 

Dividing  by  4, 

x^       225 

And 

• 
•  • 

4:^              60,     ) 

Let 

X  - 

number  of  yards. 

Then         40^  -i 

r  X 

price. 

40J- 

'-X 

81 

2X 

A      A                        81 

And              — 

2X 

3  '  54 

Mult.,  etc., 

X^ 

729. 

(Art.  378.) 

• 
•    • 

X 

27  yards;       ) 

And 

81 

I1.50, 

price,  i 

AFFECTED     QUADRATICS.  127 

10.  Let  X  =  number. 

Then  ^- 12  =  i8o 

4 
Transposing,  etc.,  x^  =  256 

,\        X  =:  ±  i6j  Ans* 

11.  Let  X  =  length  of  side. 
Then                   x^  =  area  of  bottom. 

6x^  =  capacity  in  cubic  feet. 

By  conditions,  6x^  = 

•^  1728 

77  77 

Cancelhng,         x^  =  -y^^-^^^ 

x^  =  (77Y,  and  2'  =  77  feet,  Ans, 

12.  Let  X  =  number. 
Then              {x  -\-  10)  {x  —  10)  =  156 

Or,  x^ —  100  r=  156 

Transposing,  x^  =  256 

i?;  =:  ±  16,  A71S. 


AFFECTED    QUADRATICS. 

First  Method,  Page  178. 

6.  Given  t,x^  —  24J"  =  —  36 
Dividing,  x^  —  ^x  ^  —12 
Completing  square,  x^  —  8a;  +  16  =  4 
Extracting  root,  ;i:  —  4  =  ±2 

2:  =  6  or  2,  yl7Z5. 

7.  Given  ^x^  —  ^ox  =  45 
Dividing  by  5,  x^  —  8rc  =  9 
Comp.  sq.,     x^ — 8a: +16  =:  9  +  16  =  25 
Extracting  root,  a*  —  4  =  ±5 

2-   =   4  ±  5 

=  9  or  —I,  J??^, 


128  AFPECTED     QUADRATICS. 

8.     Given  x^  —  dax  =  d 

Oomp.  sq. ,      x^  —  6ax  -\-  ga^  ^n  d  -{-  ga^ 


lo. 


II. 


Extracting  root,  x  —  $a  ^  ±^\/d-{-ga^ 


^  =  S^t dz  V d  +  ga^,  Ans. 


Given 

2X^  —  22a;  —  120 

Dividing  by  2, 

a;2  —  iia;  —  60 

Comp.  sq.,     x^  — 

iia:+  121  :=  60+  ^r  =  H"- 

Extracting  root, 

rv.               II                    -4-19 

^2               ±2 

/».           II   _4_   19 
•  •             •*'    '2"          ~2~ 

15  or  —  4,  Ans, 

Given 

x^  —  1 40        i2,x 

Transposing, 

x^  —  132;        140 

Comp.  sq.,     x^  — 

13^  +  ^   =    140   +  1J9    ^    72, 

Extracting  root, 

x-i^-  +-V- 

.  /v.             13      1      27 
•  •              .</    2"             2~ 

=  20  or  —  7,  A71S. 


Second  3Iethod,  Page  179. 

2,     Given  ^x^  —  gx  —  3  =  207 

Transposing,  etc.,      x^  —  $x  =  70 


Writing  value  by  rule,        i^^  =  f  ±  V70  +  | 


Eeducing, 

/y             3    _1_  a/289              3      I      I  7 
^             2    ±    V       ^                 2    It  "2" 

_  10  or  —  7,  Ans. 

Given             42:^ 

-- 

122;  +  5  —  45 

Dividing  by  4, 

x^  +  ^x        10 

By  the  rule, 

a;        -f±Vio  +  J 

Reducing, 

^  -      1  +  V-V- 

^   —    —  "2   ih  "2 

=  2  or  —  5,  ^/^^. 


AFFECTEIJ     QUADRATICS.  129 


4. 

Given           3^2  _ 
Dividing  by  3, 
By  the  rule, 
Eeducing, 

Given 

Dividing  by  4, 
By  the  rule. 

14:^  +  15        0 
x'      -\^x  -       5 

5. 

X       ]+V-5+^ 

X        }-hVt 

X       l±l       sovii 

4X^  —  gx        28 
x^       Ix  -  7 

X       |±V7  +  H 

—    9    _La/5  2  9 

—  -g^  ±  V  -^4- 
^  =  f  ±  ¥  =  4or  — ij. 

^.  ic  +  2  2a; 

6.  Given  1 —  =  2 

2.7;  2^+2 

Clearing  of  fractions,     x^  +  4.x'  -|-  4  +  42-2  =  42:2  _|_  g^ 
Eeducing,  x^  —  40;  +  4  =  o 

Extracting  root,  a;  —  2  =  o 

, ,  X  ^^^   2,    JLll/St 

7.  Given    x^  -\-  ~  —  ab  •=  d 


«2 


Transposing,  etc.,    2;= 7±r«^  +  f^+ -72? ^^^^' 

8.     Given  x"^  +  4«.t  =  J 


By  the  rule,  x  =:  —  2a  ±Vb  +  40,%  Atis. 

9.     Given  ^x^  —  74  =:  6:/;  +  31 

Transposing,     ^:^  —  6x  ^  105 
Dividing  by  3,    :6-2  —  22:  =1  35 

X  =  I  ±  V35  H-  I  =  I  ±^36 
=  I  ±  6  =  7  or  —  5,  ^/i5. 

10.     Given  x^  -{-  13  =z  6x 

Transposing,      x^  —  6x  =z  —  13 
By  the  rule, 
Eeducing, 


X 

^+V- 

-13  +  9 

X 

,.+V- 

-4 

X 

— 

3  +^'T 

T-0 

—  3  ±  2\/—  I,  Am, 


130  AFFECTED     QUADRATICS. 

11.  Given  (x  —  2)  (x  —  i)  =  20 
Or  x^  —  3a;  +  2  =  20 
Transposing,       x^  —  3^  =  18 

By  the  rule,  ^  =  J  ±  Vi8H-|  =  J±  V^ 

ic  m  J  ^fc  f  =  6  or  —  3,  Ans. 

12.  Given  ^-  H =  -y^ 

Clear,  of  frac,  6x^-{-i2x-{-6-\-6x^  =  i^x^  +  13:?; 
Transposing,  etc.,  x^  -{-  x  z=  6 


By  the  rule,  ^  =  —  i  ±  V6  +  :J: 

Reducing,  x  =  —  i  ±  f 

=  2  or  —  3,  ^ws. 

13.     Given    a;^ \-  c7i  ^  hd 

c 

h         /  W 

Transposing,  etc.,    x  =z  —  ^^y  hd  —  ch -\ — ^^,  Ans, 


Third  Method,  rage  181. 

3.  Given  3^^  +  4a;  =  39 
Completing  sq.,     ^x^  +  i2cc  +  4  =  121 
Extracting  root,  3^  +  2  =  ±11 

x  —  2,  or  —  4 J,  Ans. 

4.  Given  a;^  —  30  =  —  x 
Transposing,  x^  -\-  x  ^  30 
Completing  sq.,  43?^  +  4X  -\-  i  =  1 20  +  i  =  121 
Extracting  root,                2a;  -f  i  =  ±11 
Transposing,  2X  =  —  i  ±  1 1 

X  =  s  or  —  6,  ^ws. 

5.  Given  ^x  -\-  $a^  =  2 
Or  3:^2  +  5.r  :=  2 

Comp.  sq.,         36:^:2  ^  5q^  -|-  25  =  24  +  25  =  49 
Extracting  root,  6x  -\-  ^  =  ±7 

Transposing,  6x  =  —  5  ±  7 

a;  =  1^  or  —  2,  ^^Z5. 


AFFECTED     Q  U  A  D  li  A  T  1  C  S  .  131 

6.  Given  43^2  —  y^  —  2  =  0 
Or  4X^  —  'jx  =z  2 

Comp.  sq.,       640:2  —  1120;  4-  49  =  32  +  49  =  81 

Extracting  root,  8x  —  7  =  ±9 

Whence,  80;  =  7  ±  9 

0;  =  2  or  —  J,  A)is. 

7.  Given  5^2  +  20:  =  88 

Completing  sq.,  25:6-2  _|_  lox-  +  i  ^  440  +  i  =441 

Extracting  root,  5^  +  i  =  ±  21 

Whence,  50;  =  —  i  ±  21 

0;  =  4  or  —  4|,  A?is. 

8.  Given  20^2  _  5^:  ==  8 

Completing  sq.,    4o;2  —  12:?; +  9  =  16  +  9  =  25 

Extracting  root,  2a;  —  3  =  ±5 

Whence,  2X  =  s  :t  5 

X  =  4  or  —  I,  A?is. 

9.  Given  30^2  +  ^x  =  42 

Comp.  sq.,         s6x^  +  600,  +  25  =  504  +25  =529 

Extracting  root,  60^  +  5  =  ±23 

Whence,  6ar  =  —  5  +  23 

x  =:  ^  or  —  4|,  Ans. 

10.  Given  o:2_  j^^^  —  _  ^^ 

Comp.  sq.,        4x'^  —  600;  +  225  =  — 216  +  225  =  9 

Extracting  root,  20:-- 15  =  ±3 

Whence,  20;  =  15  ±  3 

a;  =r  9  or  6,  ^^^5. 

11.  Given  gx^  —  jx  =z  116 

Comp.  sq.,     3240:2  —  2520:  +  49  =  41764-49  =  4225 
Extracting  root,  180; — 7  =  ±65 

Whence,  180'  =  7  ±  65 

0:  =  4  or  —  3f,  Ans. 


132  AFFECTED     Q  U  ADU  ATIC  S. 


EXAMPLES. 

rage  182. 

I.     Given  x^  —  ^x  =  —  3 


Completing  square,  < 
Reducing, 

etc.,           X       2  ±  V—  3  +  4 
X         2  ±  I 
X       3  or  1,  A71S. 

2. 

Given 

Completing  square, 
Extracting  root. 

x^  —  Sx       —  4 

^  -  1  +  V-  4  +  ¥ 

•^          2  It  2 

X  —  4  or  I,  Ans. 

3- 

Given 

Comp.  sq.,        16.^2 . 

Extracting  root, 

Whence, 

2X^  —  ^X           —  3 

-S6x  +  49  —  —  24  +  49  _  25 

4-^'—  7  —  i:  5 

4^  -  7  =:  5 
X       3  or  i,  A71S. 

4. 

Given 

Completing  square. 
Reducing, 

x^  +  lox        24 

etc.,          X       —  5  ±  1/24  +  25 

i^       — 5  ±  7 

2;  .      2  or  —  12,  ^W6'. 

5- 

Given                   6x^ 

Or, 

Dividing  by  6, 

By  2d  method. 

Reducing, 

Or, 

'  —  132:  +  6        0 
6x^  —  132;        —  6 

a;2  _  l^-x          —  I 

X       ii±  V^'^ 

^             13    _| S_ 

•^   —   T  2           T2 

a,        i^  or  f.  Alls, 

6. 

Given 

Changing  signs, 
By  2d  method. 

\4X  —  x^       33 
a;2       1 4.1;  _       33 

.T        7  ±  V  — 33  +  49 

a:        7+4 

11  or  3,   Ans, 

ATFECTEU     CiUADRATlCS.  133 

7.  Given'  a;^  —  3  =  '—7 — 

Miiltii)lyiiig  by  6,         6x-  —  iS  =z  x  —  $ 

Transposing,  6a;"^  —  x  =  15 

By  3(1  method,    1440:2 — 242;+ i  =  360  -|-  i  =  361 

Extracting  root,  i2X  —  i  =  ±  ^9 

Whence,  12a;  =  i  ±  19 

X  =^  1 1  or  —  1 2,  vl??5. 

8.  Given  ^^ — 

7  5  140 

Mult,  by  140,         ioo:?;2  _}_  i^^^^:  =  —  73 

By  3(1  method, 

100002^  4-  196002;  +  (98)2  =  —  7300  +  9604 

=  2304 

Extracting  root,         1002  +  98  r=  ±48 
Whence  1002  =  —98  +  48 

^.                            16        100  —  OX 
o.     Given  ^ —  =  3 

^  X  ^x^ 

Clearing  of  f rac,  642 —  1 00  +  92'  ==  1 2x^ 

Transposing,  122^  —  732  =  —  100 

By  3d  method, 

576^2  _  22042'  -f-  (73)2  =  —  4800  +  5329 

=  529 
Extracting  root,  240;  —  73  =  ±  23 

Whence,  24.?;  =  73  ±  23 

2  =  4  or  2^,  Ans. 

^.  a      X       2 

10.     Given  — \-  -  z=z  ~ 

X      a       a 

Mult,  by  ax,  a^  -\-  x^  =  2X 

Transposing,  x^  —  2x  ^  —  a^ 


By  2d  method,  x  =  i  ±,  V —ci^+i,   Ans. 


134  AFFECTED     QUADRATICS. 

11.  Given      ;/~  +  2mx  =  l~  • 

By  2d  method,     a;  =  —  iii  ±:  ^/J?  +  m^,  Ans, 

X 

12.  Giveii  x^  -\-  ~  =z  il 

o 


By  2d  method,     x  =  —  tV  ±  Vl  +  zh 

16    m    V   256 

Reducing,  x  =^  —  ji^-  +  || 

2;  =  I  or  —  i-J,  An&, 

,^.                            x^  —  lOit'^  4-  I 
13.     uiveu  — ^ z=z  X  —  3 

Mult,  by  denom.,  x^—iox^-\-i  =  a:^— 92:2-1-272;— 27 
Transposing,  a^  -\-  272;  =  28 


By  2d  method. 
Reducing, 

ItIVPTI                          -     - 

X 

• 
•  • 

—  I 

X—          -y±V28-f7|9 

2  7_     1      V  8  4  I 
2     ^     V       4 

^                     2  7      129 
J.                      2^2 

.7;       I  or  —  28,  Ans, 

gx-\-  7 

14. 

14  —  2;  32;  X 

Clearing  of  fractions, 

i2a;2 — 142:  +  2;^ -f  14 — a;  =  378a; — 2  7^:2 -{-294 — 2iiC 
Transposing,  402;2— 3722;  =z  280 
By  3d  method, 

4002;'^  —  37202;  +  (93)^  =  2800  -f  8649  =  1 1449 
Extracting  root,  202; — 93  =  ±  107 
Whence,  202:  =  93  +  107 

X  ==  10  or  —  ^,  Ans. 

15.     Given  2\/^^  —  42;  —  i  =  —  42; 

Transposing,  2/^/2:2  _  4:^  =  i  —  42; 

Squaring,  42;^  _  162;  =  i  —  82;  +  162:^ 

Transposing,  1 22:^  +  82:  =  —  i 

Dividing  by  12,  2;^  -f  -^-^x  =  —  ^ 

By  2d  method,  x  =  — iiV— i^-fj 

=  -  i  ±  vs 

•*•         X  =^  —  J  i  ? 

■i  or  -2 


=z  —  i  or  —  L  ^4;zs. 


AFFECTED     QUADRATICS.  135 

i6.  Given  ^/x  +  5+6  =  a; +  5 
Tmnsposiiig,  ^/x  +  5  ^^  x  —  \ 
Squaring,  a;  -f  5  =  x-  —  2X  +  i 

Transposing,     x^  —  $x  =z  4 


By  2(1  method,  ^  =  I  ±  A/4  +  I  =  f  ± 

=  4  or  —  I,  ^y^5. 

SECOND  SOLUTION. 

Vx  -{-  s  +^  =  ^-i-  5 


Transposing,  x  -{-  $  —  Vx  +  5  =  6 
By  2(1  nietliod,  Vx  +  5  =:  i  ±  a/6  +  | 

Reducing,  V^J  +  5  =  i  ±  2 

Uniting,  \/x  +  5  =  3  or  —  2 

Squaring,  a;  +  5  =  9  or  4 

X  =z  4  or  —  I,  A71S. 

Note. — In  tlie  above  solution,  a;  +  5  is  regarded  as  a  simple 
unknown  quantity,  whose  value  is  to  be  found.  For  beginners,  it 
might  be  more  intelligible  to  substitute  y  for  it.  Then  the  equation 
would  he  y  —  \/y  =  6,  which  can  be  readily  solved  as  above. 

17.  Given  ^x^  —  jx  —  20  =:  o 
Or,  ^x^  —  7a;  =  20 

By  3d  method,    36.1:2—84^^  +  49  =  240  +  49  =  289 
Extracting  root,  6x  —  y  =  ±17 

Whence,  6x  =  7  ±  17 

X  =  4  or  —  if,  Ans. 

18.  Given  "jx^ —  160  =  3a; 
Transposing,  yx^ —  3X  =  160 

By  3d  method,    196.1-2 — 84a: +  9  =  4480  +  9  =  4489 
Extracting  root,  H-'*^'  —  3  =  ±  67 

Whence,  14^  =  3  ±  67 

a:  =  5  or  —  4f,  Ans., 


13G  AFFECTED     QUADRATICS. 

19. 


20. 


21. 


22. 


23- 


24. 


Given 

2Jl?  —  2X           l\ 

Multiplying  by  z, 

4X^  —  4X       3 

By  3cl  method,       4:z;2 

—  4:2^+1        4 

Extracting  i-oot, 

2X —   I             --2 

Whence, 

2X           I    ±  2 

X        I J  or  —  1,  A71S. 

Given                 (x  — 

2)  (x  —  i)        6 

Eeducing, 

x^  —  SX  —  4 

By  2d  method, 

X       f  +  V4  +  1 

Or, 

ry>                 3     _|_     5 

•*'    —   ^  TV.   2 

*. 

X  —  4  or  —  I,  Ans. 

Given 

4  {x^  —  i)        4X  —  I 

Or, 

42;2  —  4a;  _  3 

(See  Ex.  19.) 

X       I J  or  —  ^,  A71S. 

Given 

(2X  —  3)2  _   SX 

Eeducing, 

4X^  —  20a;        —  9 

By  3d  method,  ^x^  — 

-  20a;  +  25        —  9+25        16 

Extracting  root. 

2X  —  S  —  -  4 

Whence, 

2x  —  s  -  4 

X  —  4^  or  J,  Ans. 

Given 

14 
3^—2       

'^                          X—  I 

Mult,  by  ^  —  I,  etc., 

^X^  —  ^X            12 

By  3d  meth.,   363^2  — 

-  602-   +   25    —    144  +   25    —    169 

Extracting  root. 

6x  —  s         +  13 

Whence, 

6x  —  s    -  13 

X       3  or  —  i|,  ^^^s. 

Given                     ^x 

14  —  .r 

: 14 

Mult,  by  2:  4-  I,  etc., 

4^2  —  g^^-           28 

By  3d  meth.,     64:^:2- 

-i44a:  +  8i        448  +  81        529 

Extrncting  root. 

8ic  —  9        H-  23 

Whence, 

8a:  —  9       23 

a;  =  4  or  —  ij,  J«s. 

AFFECTED     QUADEATICS.  137 

25.  Given  x^  4-  ~  =  ^- 

25         5 

Transposinsr,  x^  —  ~zzz  — ^ 

5  25   ^^ 

By  2d  method,  x  =  f  ±  V— AT^ 

Reducing,  a;  =  f  ±  | 

a;  =  -|  or  I,  ^4?25. 

26.  Given        x^  -{-  -  =  - 

2        2 

By  2d  method,  x  =  -i±  ViT^  =  -  i  ±  V^ 
Reducing,  ^  =  —I  ±i 

X  =z  i  or  ~  I,  A71S. 

27.  Given  x^  —  2nx  =  tv^  —  n^ 

By  2d  method,  x  =  n  ±_  Vrn^  —  n^  +  tv^ 

,;        X  =.  n  ±:  m,  A  ns. 

28.  Given  9(i-^^x-ia 

X  if 

Clearing  of  fractions,  9^2  _  g^^  =  x^  ^  ^ax 
Transposing,  x^  —  ^ax  =  9^  —  c^nh 


By  2d  method,     x  —  ^±  \J  ^h'^  _  ^ah  +  ^ 
Reducing,  ^  =  f  ±  ^g:Z36^+^^ 

Or,  i.  =  ^  ±  ^^'  -  3^ 

2  -^        2 

X  =1  ih  or  3«  —  35,  ^^. 


138  AFFECTED     QUADRATICS. 

EQUATIONS    OF    THE    QUADRATIC    FORM. 

Fiifje  183. 


4. 

Given 

a:^  +  8  =  6i«;2 

Transposing, 
By  2d  method, 

^- 

6a;2  —  —  8 

x^  _  3  -  V-  8-9 

Reducing, 

x^       3  ±  I 

Uniting, 

a^  —  4  or  2 

Extracting  root, 

X  —  i  2  or  --  V2,  A71S. 

5- 

Given             od^  - 
By  2d  method, 

-  2X^ 
X^ 

-  3 

—  I  -  V3  +  I 

Reducing, 

X^ 

1  +  2 

Uniting, 

X^ 

=  3  or  —  I 

Extracting  root, 

X 

=  ±  V3  or  -f-  V—  I,  Ans. 

6. 

Given 

x^  — 

yx^  -.  0 

Transposing, 

x^       ja^ 

Dividing  by  x^. 

a?  =z  y 

Extracting  cube  root. 

X       Vl        i«9i4-,  Ans. 

7. 

Given 

2  +4       ^' 

Multiplied  by  2, 
By  2d  method. 

^^\-i. 

X           -i±VA  +  TV 

Reducing, 

/v.                      I    _U    2 

X  —  \  or  —  J,  Ans. 

8. 

Given 

\/x^  +  i\/x  =  I 

By  2d  method, 

v^-  -i±Vi  +  T% 

Reducing, 

^:r  =  -  1  +  i 

Uniting, 

Vii^       i  or  —  2 

Involving, 

X       \  or  —  8,  ^1??*', 

AFFECTED     QUADRATICS.  130 

9.     Given  4X  -{-  a^^/x  +  2  =  7 

Transposing,  4V^'  +  2  =  7  —  4a; 

Squaring,  16a;  +  32  =  49  — 562:+ i62;2 

Transposing,  162;^ —  72::?;  =  —  17 

By  3d  me.,    2562'^ — ii52a;-|-(36)^  =  — 272+1296 

^  1024 
Extracting  root,  16a:  —  36  =  ±  32 

Whence,  16a;  =  36  +  32 

a;  =  4I:  or  J,  ^;^5. 

^.                                     V4^+2o        4  _  V^ 
10.     Given  —  — ~— 

4  +  \x  vx 

Clearing  of  fractions  and  squaring, 

^x^  +  20X'  =  256  —  32:?;  +  x^ 
Transposing,  ^x^ -\- ^2X  =  256 

By  3d  meth.,    92:2+ 156.^+ (26)2  r=  768  +  676=1444 
Extracting  root,  30;  +  26  =  ±38 

Whence,  3X  =  — 26  +  38 

a;  =  4  or  —21  J,  Alls. 

PROBLEMS. 
Page  184, 

1.  Let  X  =  One, 
Then                          12  —  x  ^  The  other. 
And                         12a;  —  a:^  =  32 

Or  x'^  —  i2x  =  —  32 

Comp.  sq.,      x^—  i2X  +  36  =  —3^  +  3^  (^I't.  335.) 
Extracting  root,  2;  —  6  =  +2 

Whence,  a::  =  8  or  4  ;  )     . 

12  —  ic  =  4  or  8,   ) 

2.  Let  X  =  Cost, 

Then  - —  =  Rate  per  cent, 

100 

And  —  =  Percentage  lost. 

joo 


140       ,  AFPECTED     QUADRATICS. 

By  conditions,         x —  =  24 

•^  100 

Clearing,  1002:  — .t^=  2400 

Changing  signs,   x^  —  looa:  =  —  2400 


By  2d  method,  x  =  50^^—2400  +  2500 

a;  =  50  ±  10 
=  $60  or  $40,  Ans. 

3.  Let  X  =  one  number, 
And  y  =  the  other. 
Then  x  -{-  y  =z  10  (i) 
And  xy  =  24  (2) 
Squaring  (i),  a^  -\-  2xy  +  y^  =  100  (3) 
Mult.  (2)  by  4,  4xy  =  96  i4) 
Sub.  (4)  from  (3),  x^  —  2xy  -^  y^  =  4'  (5) 
Extracting  root,  x  —  ?/  =  ±2  (6) 
Bringing  down  (i)  x  +  y  ^  10 

Adding  (6)  and  (i),  etc.,  a;  ==  6  or  4;  | 

Subtracting  (6)  from  (i),  etc.,       y  =  4  or  6,  f 

Note.  — Let  tlie  pupil  compare  this  solution  with  that  of  Problem  i, 
in  which  one  unknown  quantity  is  used. 

4.  Let  X  =  no.  of  sheep. 
Then                                             —  ==  price  of  each. 

X 

80          80 
And  —  = h  I 

X         X  -{-  4 

Clearing  of  fractions,      80a:  +  320  =  80:2;  -{-  x^  -^4X 

Rejecting  802:  from  each,    x^  +  43;  =  320 

Completing  square,      x^  +  4X  +  4  —  324 

Extracting  root,  x  -{-  2  =  ±18 

Whence,  x  =  —  2  ±  18 

X  =  1 6  sheep] 

AT  80        „  ,    \Ans, 

And  —  =  $5  each,  J 

X         ^  ' 

The  problem  will  not  admit  of  a  negative  result.     There- 
fprQ  the  Ans.  16  sheep,  at  $5  a  head. 


AFFECTED  QUADRATICS.                        141 

Let  X  =:  the  number. 

Then  2x^  ^65  —  32' 

Transposing,  2x^  +  3X  =  65 

Dividing  ])y  2,  x'^  -\-  ^x  zzz  ^ 


Completing  square,  etc.,  a;  =  —  J  ±  V-^^  +Ti 


TF 


r^    3    +    23 


=  5  or  —  6 J,  A71S. 

6.  Let  X  =  No.  of  scholars, 

Then  =    "    "  oranges  each  received. 

X  ^ 

A    1  144         144 

And  -—  =  -^^-  +  I 

X         x  -\-  2 

Clearing  of  frac,        144a;  +  288  =  1443;  +  x^  -\-  2X 

Or  a?  +  2x  =  288 

By  2d  method,  x  z=  —i  ±^  A/288  +  i 

a;  =  —  I  2b  17 
=  16  or  —  18. 

Note. — The  second  value  of  x,  being  negative,  cannot  be  applied  to 
children.  It  may  be  observed  that  negative  values  do  not  apply  to 
concrete  numbers.     Hence  the  Ans.  16  scliolars. 

7.  Let  X  =  share  of  one, 
Then                         50  —  .t  =      "      "  the  other. 
And                       50.T  —  x^  =^  600 

Changing  signs,    x^  —  50.^  =  —  600 

By  2d  method,  x  =  2^  ±:  V—  600  +  625 

X  =  830,  one;    \^^^^^ 
And  ^o  —  X  =z  $20,  other,  )  ^ 

8.  Let  X  =  one, 
Then                       100  —  x  =  the  other. 
And                       loo:?;  —  a;^  =  2400 
Changing  signs,    a^ — loox  =  —  2400 


By  2d  method,  x  =  SoiV — 2400  +  2500 

.T  =  60  ;  )     , 
Ana  100  —  X  =  40,  ) 


142  AFFECTED     QUADRATICS. 

9.  Half  the  perimeter  (i|-^  =  64),  or  64  rods,  is  equal  to 
the  sirm  of  Ibe  length  and  breadth. 

Let  X  =  the  length, 

Then  64  —  x  =^    "   breadth. 

And  64X  —  x^  =^  1008  sq.  rds.,  area. 

Changing  signs,  x^  —  64X  =  —  1008 


By  2d  method,  x  =  32^^—1008+1024 

X  =  36  rds.,  I'gth ;    ) 
64-x  =  2S    "    brmh,  j 

10.     Let  X  =  number  in  file. 

Then  ic  +  60  =       "         "  rank. 

And  x^  +  60a;  =  1600  men. 


By  2d  method,  x  =z  —  30  ±  a/i6oo  +  900 

Keducing,  x  =  —30  +  50 

a;  =  20  men  in  file;    )    . 


X 


+  60  =  80    "     "  rank,  f 


Page  185, 

II.     Let  X  ~  number  of  lambs. 

Then  -—  =  cost  of  each. 

X 
I TX 

And  s\'^>    ^^    —  =  ^™t*  received  for  them. 

By  conditions,  50  =  — 

-'  '  2  X 

Clearing  of  frac,     \ix^  —  loo^r  =100 
By  3d  method, 

4842:^  —  44oo:r  +  1 0000  =  14400.  (Art.  ^^6.) 
Extracting  root,         22^  —  100  =  ±120 
Transposing  and  dividing,       a;  =  10  lambs,  Ans. 


AFFECTED     QUADRATICS.  143 

12.     het  ^  ^  one  number, 

'X'hexi  ^  —  X  —  the  other. 

Clearing  of  frac,  etc.,     x^  —  4X  =  —  4 
By  second  method,  x  =  2  ;  |.  ^^^ 

And  4  —  a;  =  2,  p 

SECOND  SOLUTION. 

Denote  the  numbers  by  x  and  y. 

Then  x -\- y  =  4  (i) 

And  -  +  -  =  I  (2) 

X      y 

Clearing  (2),  x  +  y  —  xy  (3) 

Subtracting  (3)  from  (i),         o  —  4  —  xy 

Or,  xy  —  4  (4) 

From  (i)  and  (4)  find  the  value  ot  x  —  y,  thus: 

x  +  y  =  4  (5) 

Squaring  ( I ),  x^-\-2xyi-y^  =  16  (6) 

Mult.  (4)  by  4,         4^y         =  16  (7) 

'Subt.  (7)  from  (6),  x^—2xy-\-y'^  =  o 
Extracting  root,  x  —  y  =  o 

Or,  x  =  y  (8) 

Substituting  in  (i),  2x  =  4 

.T  =  2 
From  (8),  y  =  2-    ^^^s.  2  and  2. 

13.     Denote  the  numbers  by  ic  and  ?/. 

Then  x  -{-  y  =  s  (i) 

And  a::^  _p  ^3  _  5^  ^2) 

Dividing  (2)  by  (i),  x^—xy-\-y'-  =  13.  (Ax.  =;.)     (3) 
Squaring  (i),  x^-{-2xy  +  y'^  =  25  (4) 

Subt.  (3)  from  (4),         zxy         =112  (5) 


144  AFFECTED     QUADRATICS 


Or, 

xy  —  4 

Whence, 

y  =  l 

(6J 

Substituting  in  (i), 

^  +  1  =  5 

Clearing,  etc.. 

a^  —  5*  —  —  4 

X      1  ±  a/- 

By  2d  method. 

-4  +  ¥ 

^'  =  I  ±  f  =  4  or  I 
Substituting  in  (6),  «/  =  i  or  4. 

A71S.  4  and  I. 

This  problem  may  be  solved  with  one  unknown  quantity, 
thus, 

Let  a;  =  one  number. 

Then  5  —  a;  =  the  other. 

And     125  — 75a: -f  15^:2 — oi?-\-x^  =  65 
Transposing,  etc.,     i^x^  —  ^^x  ■=  —  60 
Or,  x^  —  ^x  =:  —  4 

Completing  sq.,  etc.,  ic  =  f±  V— 4+^- 

a:  =  4  or  i 
And  5  —  .T  =  I  or  4 

Ans.  4  and  i. 

i4.     Let  ir  =  No.  of  yards  in  the  width. 

Then  a;  +  i  =    "        "  "        length. 

I  acre  =    4840  square  yards. 
3  acres  =  14520      "         " 
Length  x  width     =  a:^  +  a:  =  14520 
By  2d  method,     x  =  —  I  ±  V14520  +  ^ 
Reducing,  x  =  —^  ±  V^-^ 

Or,  x=  -i±  ^fi 

X  =  1 20  yards,  width ;  )     . 
X  -\-  I  =  121      "      length,  f 


AFFECTED     QUADRATICS.  145 


15.     Let  X  =  B's  rate, 

Then.  x  +  i  —  A's     " 


- —  =  time  it  takes  B  to  go. 


300 

X  -\-  1 


a  a  A       <.' 


And  3£?  ^  ^o^ 

X  X  -\-  1 

Dividing  liy  lo,  clearing,  etc., 

.1-2  -j-  ic  =  30 
By  2(1  metli.,       x  ^=  —  |  +  V30  +  J 
Reducing,  2;  =  —  }  ib  V" 

a;  =  5  miles,  B's  rate ; 
And  X  -\-  I  =  6     "      A' 


s     " 


Ans. 


16.     Let  X  =  No.  that  B  relieves, 

Then  a;  +  40  =  "       "A      " 

=  sum  B  gives  to  each, 

"     A     "        " 


X 

1200 
a;  +  40 


T,           ,.,.  1200          1200 

By  conditions,  = h  5 

•^                    '  X          a-  +  40      ^ 

-rx'     •  T            1  240                   240 

Dividmor  by  t:  -^^  = \-  1 

Clear,  of  frac,  240.^  +  9600  ^  240.^  +  x-  +  40.T 

Transposing,  a;^  -|-  40a:  =  9600 


By  2d  method,  ic  =  —  20  ±  1/9600  +  400 

Reducing,  x.  =  —20  +  100 

X  =  80 
AVhence,  ic  +  40  =  120 

80  relieved  by  B ;  )    . 
And  120       "        "  A,  ) 


146  AFFECTED     QUADPtATICS. 

17.     Let  X  =:  one  part. 

Then  48  —  ir  =  the  other. 

And  x^  —  48:r  =  — 252 

Completing  square,  etc.,  Ans.  42  and  6. 


BY  TWO   UNKNOWN   QUANTITIES. 

Denote  the  parts  by  a;  and  y. 

Then 

X  -\-y  —  4S 

(I) 

xy  —  252 

(2) 

Squaring  (i), 

{x  +  yf  —  2304 

(3) 

Multiplying  (2)  by 

4,           /^xy       1008 

(4) 

Subtracting, 

{x  —  y)^        1296 

(5) 

Extracting  root. 

X  —  y        ±36 

(6) 

X  -\-  y       48 

(0 

Adding  (i)  to  (6), 
Subt.  (6)  from  (i). 

etc.,            X       42  or  6 ;  )    . 
etc.,            y       6  or  42.  ) 

ns. 

i8.     Let 

X       number  A  bought, 

And 

y  -      "        B       -^ 

Then                      i 

X  +  y  —  10 

(i) 

12                                                  A                  •    T                • 

—        cents  A  paid  apiece. 

X 

(^) 

12 

y 

(3) 

By  conditions. 

12     12 

—  —  —  +  I 

X         y 

(4) 

Clearing  of  frac, 

i2y        12.1;  -|-  a:?/ 

(5) 

Taking  value  of  x  from  (i), 

12?/  =  12(10—//)  f  y(io-y) 
Eeducing,  i2y  z=  120  —  127/  +  10^  —  y^ 

Transposing,     y"^  +  14?/  =120 
By  2d  method,  y  ^  —  7  ±  ^169 

Eeducing,  y  =  —  7  ±  13 

y  =z  6  lemons,  B ;  ^    . 
From  (i),  X  =  4        "      A,  J 


AFFECTED     QUADEATICS.  14t 

BY   ONE  UNKNOWN    QUANTITY. 

Let  X  =  No.  A  bought, 

Then  ^o  —  x  =    "     B       " 

_  _    .  12  12 

By  conditions,  —  = h  i 

•^  ic         lo  —  X 

Clearing  of  frac,  etc.,    x^—$4x  =  —  120 


By  2d  method,  x  =  ly  ±  V—  120  +289 

a;  =  17  +  13  =  4,  A;  J 
And  10  —  X  =  6,  B,  ) 

19.     Let  ^  =  the  breadth. 

Then  24  —  .t  =  the  length. 

By  conditions,       24X  —  x^  —  35(24—  2x) 
Eeducing,  x^  —  94a!;  =  —  840,  etc. 

A71S.   14  ft.  length;  10  ft.  breadth. 

BY  TWO  UNKNOWN  QUANTITIES. 

Let  X  =  the  length, 

And  y  =  the  breadth. 

Then     x  -\-  y  =  24  ft.,  or  half  the  perimeter.       (i) 
xy  =  35  (a;  —  y),  the  area.  (2) 

Mult,  (i)  by  35,     35^  +  352/  =  ^4°  (3) 

Transposing  (2),    35-^—35^  —  ^y  (4) 

Subt.  (4)  from  (3),  70?/  =  840  —  xy 

Takingval.  of  a:from(i),  joy  =  840  —  (24  —  y)  y 
Eeducing,  Toy  =  840  —  24^  +  y^ 

Transposing,  etc.,    y^—g^y  =  —  840 
By  2d  method,  ^  =  47  ±  V— 840  +  2209 

^  =  47  ±37 

y  =  10,  breadth  ;  ^   ^^ 


And  X  =  14,  length 


^Am 


148  AFFECTED     Q U  A D  II ATICS  . 

20.  Let  X  ■=.  No.  of  rows. 

Then  a;  +  3  =    "      "  trees  in  each  row. 

And  a;2  +  3:^  =  1 80 

By  2d  method,  x  =:  —  J  ±  V180  +  \ 

Eeducing,  ^  =  —  f  ±  ¥ 

OJ  =  12  rows  ;  l  4    . 

And  ic  +  3  =  15  trees  in  each,  j  " 

21.  Let  a;  =  digit  in  tens'  place, 
Then                       7  —:?;=:     "       "  units'    " 

And  i^^  +  (7  —  "x^^  =  29 

Eeducing,  x9-  —  7a;  =  —  10 

Completing  square,  etc.,  Arts,  52. 

BY  TWO  UNKNOWN   QUANTITIES. 

Let  loa;  +  «/  =  number. 

Then  a;  +  ?/  =  7  (i) 

And  a;2  4-  ^2  _  29  (2) 

Squaring  (i),        x^  +  22:?/  +  «/2  z=  49  (3) 

Subt.  (2)  from  (3),       2xy  =  20  (4) 

Combining  (2)  and  (4),    x  —  y  =    $  (5) 

(5)    "     (0.  ^=    5 

y  =    2 

(See  solution  of  Ex.  17.)         iox-}-y  =  52,  Ans. 

22.  Let  x  =  No.  in  the  party, 

Then        x  +  ^o  z=  contribution  required  of  each. 

And       x^  -\-  ^ox  =z  1000 

By  2d  method,  x  =  —15^^/1000  +  225 

Keducing,  --^  =  —  15  ±  35 

a;  =  20  persons,  Ans. 


AFFECTED     QUADRATICS'.  149 


Page 

186. 

23- 

Let 

X 

— 

the  less, 

Then 

* 

I20 
X 

— 

"    greater. 

And          (x 

— 

'   \   X 

-4 

120 

Reducing, 

x^ 

+  2X 

8o,  etc. 
Ans,  8  ant. 

15- 

BY  TWO  UNKNOWN  QUANTITIES. 

Let  X  =  the  greater,  and  y  =:  the  less. 

Then  xy  =  120  (i) 

And               (y  +  2)  {:c  —  3)  =  120  (2) 

From  (2),      xy-^2x—^y  —  6  =  120  (3) 

From  (i),    xy  ^  120,  and  a;  = (4) 

From  (3),      120  +  ^  —  3?/  =  126  (5) 

Clearing,  etc.,  3?/^  -f  6?/  ==  240 

Dividing  by  3,  y^  -\-  2y  =z  80 

By  2d  method,  «/  =  —  i  ±  V^ 

Reducing,  ?/  =  —1+9 

7/  =    8  or  —  10.  less  ;       )   j    , 
From  (i),  a;  =  15  or  —  12,  greater,  f 

24.     Let  x  =  one, 

Then  36  —  x  =  other. 

By  the  conditions,  ;^6x  —  x'^  =  80  (36  —  2x) 
Reducing,  x^  —  ig6x  =  —  2880 


Completing  sq.,  etc.,  x  =  98  +  ^—2880  +  9604 

Or  X  =  g8±V6'j24 

And  a:  =  98  ±  82 

X  =  16  :  ) 
And  36  —  a"  =r  20,  3  ^ 


150  AFFECTED     QUADRATICS. 

25.  Denote  the  numbers  by  x  and  y. 

Then  x -\- y  =  ^s  (0 

And  xy  \  x^  -\-  y'^  \:  2  '.  ^  (2) 

Changing  (2)  to  an  equation, 

2X^  +  2«/2  =  ^xy  (3) 

Squaring  (i),  x^  +  2xy  -{- y^  =  5625  (4) 

Transposing  and  mult,  22:2+2?/^=  11 250 — 4xy  (5) 
Equating  (3)  and  (5),  s^y  —  1 1250— 42:2/  (6) 

Reducing,  xy  =  1250  (7) 

Combining  (4)  and  (7),      x  —  ?/  =  25  (8) 

Combining  (i)  and  (8),  x  =  ^o  ;  )     . 

Substituting  in  (8),  ?/  =  25,   f 

BY  ONE  UNKNOWN  QUANTITY. 

Let  X  rr:  oue  number, 

Then  75  —  a;  =  the  other. 

And      75rc  —  x^  \  x^  -\-  (75  —  .t)^  : :  2  :  5 

Placing  the  product  of  the  extremes  equal  to  the  product 
of  the  means  and  reducing, 

x^  —  75a;  =r  —  1250,  etc. 

26.  Denote  the  numbers  by  x  and  146  —  x. 

Then         V146  —  x  —  \/x  z=  6 

Transposing  and  involving, 

1 46  —  :?;  =  36  +  1 2  Vx  +  a? 

Transposing  and  dividing, 

s^  —X  =  e^/x 

Squaring,  3025  — 11  ox +  a.'^  =  2>^x 
Transposing,        x?' —  146.?'  =r  —  3025 
Completing  sq.,  etc.,         x  =  73±V— 3025  +  5329 
Reducing,  •'^  =  73  ±  48 

.T  =  121  or  2c;  ;  )    , 
And  146  —  X  =  25  or  121, 


AFFECTED     QUADRATICS.  151 

27.     Let  X  =  circumference  of  fore-wheel, 

And  y  —  "  "  hind     " 

mi  '?600  3600  ^  ,     ^ 

Then —  ^ \-  60     (i) 

X  y  ^  ' 

And  — = h  40  (2) 

Dividing  (i)  by  60,  -^  =  —  +  i 

Clearing  of  fractions,  Soy  =  602^  +  xy 

Factoring  and  dividing,  x  =  - — ^  (3) 


Dividing  (2)  by  40,  —^ —  =  — 1-  i 


60  -i-y 
90 
^  +  3       y  +  3 


Clearing  of  fractions, 

90^  +  270  =  gox-^27o-\-xy-\-sx-{-sy  +  g 
Transposing,  etc.,  ^ly  —  9  =  93^  +  ^y 

Factoring  and  dividing,  x  =  — ~ (4) 

Equating  (3)  and  (4),        ^    °"^     ==  -^- — ^ 

Clearing,       558o?/  +  6o^^  =:  5220^—540  +  87^/2—9?/ 

Transposing,  etc.,      27?/^  —  369/y  =  540 

By  3d  method, 

4X272?/2— 4x  27  X  369^  +  369^  =  540  X  108  +  136161 

Extracting  root,    54?/ —  369  =  ±  Vi9448i  =  d:44i 

Whence,  54^  =  810 

?/  =  15  ft.;  ) 

T^         .  ,  60  X  15       60  X 15  ,,    \  Ans. 

From  (3  ,  .?;  =  :; = =  12"     \ 

^^  60+15  75  ) 

BY  ONE  UNKNOWN   QUANTITY. 

Let  X  =  revolutions  of  hind-wheel, 

Then  60  -\-  x  z=z         "  "  fore-wheel, 

And =  circumference  of  hind-wheel, 

x 

And  J^^^  _  ,,  u  fore-wheel. 

69  -^  X 


152  AFFECTED     QUADRATICS. 

T»         XI  J'^-  3600  3600 

By  the  conditions,  -r^ =  —7 —  40 

Clearing  of  frac,  etc.,    0^24-2460:?;  =  648000 


By  2d  method,      x  =  —1230  +  ^648000  +  1512900 
Reducing,  x  =  —  1230  ±  1470 

:?;  =:  240 

^ =:  15  ft. hind-wheel; 

}  Ans. 
=  1 2  ft.  fore-wheel. 


60  +  iC 


28.     Let  X  =  one  number. 

And  X  -\-  16  =  the  other. 

By  the  conditions,      x^  -\-  i6x  =z  ^6 


Completing  square,  etc.,         x  ^=  —  8  ±  ^36  +  64 

X  ^=  2  or  — 18;  )     . 
And  a:  +  16  =  18  or  —  2,  i 

29.     Denote  the  numbers  by  x  and  y. 


Then                                 x  -\-  y 

ii 

4 

(I) 

And                                 -  +  -  - 

a;      ?/ 

3^ 

¥ 

(2) 

Multiplying  (2)  by  xy,    x -^  y  — 

i(>.nj 
5 

(3) 

I  Oxn 
Equating  (i)  and  (3), 

f 

Multiplying  by  j,                4xy 

5 

(4) 

Squaring  (1),      x^  -\-  2xy  +  if 

16 
9" 

(5) 

Subt.  (4)  from  (5),  etc.,     x—y 

I 

(6) 

Combinine^  (1)  and  (6),          x 

f:> 

And  .        y  =  \,  ) 


AFFECTED     Q  U  A  D  K  A  T  I  C  S .  *        153 

BY  ONE  UNKNOWN  QUANTITY. 

Let  X  =  one  number, 

Then  f  —  ^  =  the  other. 


And  -  +  —^—  =  ¥ 

X      4  —  3^ 

Clear,  of  frac,  etc.,    1 2^:^—  1 6x  ==  —  5,  etc. 

30.     Let  X  =  the  less  number. 

Then  x  -\-  15  =   "    greater  " 

And  ^_i5f  ^  ^ 


2 


Clearing  of  frac,  etc.,      a^+  15  =  2X^ 
Transposing,  etc.,         x^  —  io;  —  ^ 
Completing  square,  etc.,         ^  =  i  ±  V^-  + 

Reducing,  a;  =  ^l  ±  V 

Or,  X  =  l±-' 


121 


I_I 
"4 

2:  =  3  less;         )    . 
And  a;  +  15  r=  18  greater,  f 

3 1 .  Let  X  =  her  age. 

Of' 

Then  -  +  yx  —12  =  0 

2 

Clearing  of  frac,  etc.,  x+zVx  =  24 

Completing  square,  etc,       Vx  =  —  i  ±  5 

Uniting,  Vx  =  4  or  —  6 

Involving,  x  =z  t6  or  $6. 

Hence  her  age  is  16,  or  36  years,  Ans. 

"Note. — The  minus  sign  must  be  placed  before  the  radical  in  the 
first  equation  when  the  second  value  is  taken. 

32.  Denote  the  length  by  x  and  the  breadth  by  1/. 
Then  2  (.^  +  ?/)  =  96  (i) 
And                              xy  =  1o(x-y)  (2) 
For  solution,  see  that  of  Problem  19,  above. 


25  8  rods,  length  \   \    a    ^ 
20    "     breadth,  ) 


X 

A's  age, 

120 
X 

B's 

a 

154  AFFECTED     QUADRATICS. 

33'     Let 
And 

By  conditions,  (x — 3)  | [-2|  =  120 

Multiplying,  etc.,      2X^  +  II4;^;  =  1202;  +360 
Transposing,  etc.,  x^ —  3-^  =  iSo 

Oonii)leting  scp,      x^  —  32;  +  f  =  1 80  +  |  =  ^  J^ 
Extracting  root,  x  —  ^  =  di  -j- 

^  =  I  ±  -Y-,  or  15  years,  A's  age; 

ii5=  8     «     B's    "     ^'^'"- 


X 


34.     Let  X  = 


No.  of  lbs.  pei^per  bought 
for  8  crowns. 


Then  -  =  price  of  pepper  per  lb. 

X 

80  X  -  = =  sum  paid  tor  ])epper. 

XX  '  ^    ^^ 

X  —  14  =  lbs.  saffron  for  26  crowns. 

26 

=.  price  of  saffron  per  lb. 


X  —  14 


^6  X =  ^^ —  =     sum  paid  for  saffron. 

X — 14      X — 14 

And 1 — —  =  188,  by  conditions. 

X         X  —  14  '' 

Clearing  of  fractions, 

640a;  —  8960  +  936:?;  =  i88ic2 —  2632.^' 

Transposing, 

i88a;2  —  4208a;  =  —  8960 
Dividing  by  4, 

47X"2  —  1052a;  =  —  2240 


By  3d,     94^^—1052  =  iV  — 2240  X  188 +  (1052)2 

Eeducing,  94a; — 1052  =  iV — 421120  +  1106704 

"  942;— 1052  =  +V685584 

"  94a; — 1052    =:  ±_  828 

Whence,  94a:  =.  1880 

a;  =  20  lbs.  p.  for  8  c,  Ans. 


SIMULTANEOUS     QUADRATICS.  155 

SIMULTANEOUS    QUADRATIC    EQUATIONS. 

Page  ISS, 

2.  Given  cr^  +  y2  -_  25  (i) 
And  X  -^  y  —  J  (2) 
Squaring  (2),  x^  +  2x1/  +  ?/2  =  49  (3) 
Snbt.  (i)  from  (3^                 2xy  =  24  (4) 

-      (4)     "     (i),x^-2xy  +  f=     I  (5) 

Extrcicting  root,                 x  —  ?/  =  ±  i  (6) 

Combining  (2)  and  (6),            ^  =  4  or  3 ;  jj^ 

And                                            ?/  =r  3  or  4,  \ 

3.  Given  x^  -\-  y'^  =  'j^  (i) 
And  X  -\-  y  :=  12  (2) 
Squaring  (2),  x^  -f-  2a:'?/  +  ?/2  =  144  (3) 
Subt.  (i)  from  {3),                  2xy  =  70  (4) 

"     (4)     "     (i),  x^-2xy^-f  =4  (5) 

Extracting  root,                x  —  y  =z  ^  2  (6) 
Bringing  down  (2),           x  -\-  y  ^  12 

Combining  (6)  and  (2),            .t  =  7  or  5  ;  |     . 

And                                             y  =z  ^  or  7,  ) 

4.  Given  x^  —  y-  =28  (i) 
And  X  —  ^  =  2  (2) 
Dividing  (i)  by  (2),        x  ^  y   z=  14  (3) 


Ans. 


Combining  (3)  and  (2),  ^  =  8; 

And  y  zzz  6, 

Given  x^  -\-  y"^  =^  244  (i) 

And  y  —  x  ^  2  (2) 

Squaring  (2),       if  —  2xy  +  ^r^  =  4  (3) 

Subt.  (3)  from  (i),  2xy  =  240  (4) 

Adding  (4)  to  (i),  x^  +  2xy  +  if  =  484  (5) 

Extracting  root,  ^  -^  !f  =^  dz  ^^  (6) 

Bringing  down  (2),       —  x  -{-  y  =z  2 

Combining  (6)  and  (2),  x  =z  10  or— 12;) 

And  y  =  i2or  —  io_,  j 


156  SIMULTANEOUS     QUADRATICS. 


Given 

2,x^  —  if  —  251 

(i) 

And 

^  +  4^  —  38 

(2) 

From  (2), 

^  —  38  —  4«/ 

Squaring, 

x^        1444—304^+16^^ 

Multiplying  by  3,        3^'-  =  4332  —  912^  +  487/2     (3) 

Substituting  in  (i),     " 
4332  — 9i2?/  +  48?/2— «/2  =  251 

Transposing,  etc., 

^yif  —  gj2y  =  —4081 
By  3d  method, 
(47)y— 47  X9I2^/  + (456)2  =  —  191807  +  207936 

=:    16129 

Ext.  root,         47?/— 456  =  ±  127 
Whence,  47j/  =  456  ±127 

y  =  12JI  or  7;  I 

From  (2),  using  2d  value  of  y.  ) 

7.     Given  Sx^  +  52/2  =728  (i) 

And  61/  —  X  =:  15  (2) 

From  (2),  X  =  6^—15  (3) 

Squaring,  x^  =  ^6f  —  iSoi/  +  225 

Multiplying  by  8,  Sx^  =  2887/2 — 1440?/  +  1800 

Substituting  in  (i), 
2887/2— 1440?/+ 1 800 +  5  7/2  =728 
Transposing,  etc., 

2932/^  —  1440^  =  —  1072 
By  3d  method, 
(293)27/2—1440x293?/  + (720)2  —  —314096  +  518400 

=  204304 
Ext.  root,.         293?/  —  720  =r  ±452 
Whence,  293?/  =  720  +  452 

y  =  40Y  i^i;) 
X  =  g,  ^  Ans. 

Dropping  2d  value  of  y  from  (3).  ) 


SIMULTANEOUS     QUADRATiC'S.  157 


Page  189. 

9.     Given  x  +  y  ^=  g  (i) 

And  a:^  +  ^3  _  ig^  (2) 

Div.  (2)  by  (i),      x^  —  xij-\-  f  —  21       ,  (3) 

Squaring  (i),       x^  -\-  2xy  -\-  y^  =.  ^i 
Subtracting,  3^2/  =  60  (4) 

And  xy  =20  (5) 

Subt.  (5)  from  (3),  x^—2xy-\-y'^  =  i  (6) 

Extracting  root,  ^  —  ^  =  ±  i  (7) 

Bringing  down  (i),  x  -}-  y  ^  g 

Combining  (7)  and  (i),  re  =  5  or  4;  ) 

And  ?/  =  4  or  5,  ) 

10.  Given  x  —  y  =  2  (i) 
And  a^  —  y^  =z  98  (2) 
Div.  (2)  by  (i),  a:2-(-a;?/  +  2/2  =  49  (3) 
Squaring  (i),  a:;^  —  2xy  -{-  y^  =  4  (4) 
Subt.  (4)  from  (3),                3a:?/  =  45 

And  xy  =15  (5) 

Adding  (5)  to  (3),  etc.,  x  -\- y  =  ±  S  (6) 

Bringing  down  (i),  x  —  y  =  2 
Combining  (6)  and  (i),  :c  =  5  or  — 3  ;  )  ^^^^^ 

And  -  y  =  s  or  —5,   f  " 

11.  Given  3^^  —  jy^  =  —  i  (i) 
And                                        4xy  =  24                     (2) 

From  (2),  a;  =  -  (3) 

Squaring  and  mult,  by  3,      ^x^  =  — ^ 

•J 

_  _  Q 

Substituting  in  (i),    — ^ 7^^  =  —  i 

Clearing,  108  —  "jy*  ■=  —  y^ 

Transposing,       •  iy^  —  y^  ~  108 

By  3d  method,  1/2  =  4  or  —  3^ 

Dropping  negative  value,         y  =z  2  \  ] 
From  (3),  a:  =  3,  j 


158  SIMULTAN-EOUS     QUADRATICS. 


12. 


Ans. 


Given  :v^  —  xy  +  if  =19  (i) 

And  xy  =  15  (2) 

Siibt.  (2)  from  (i),   x^  —  2xy  -f  y^  =4  (3) 

Extracting  root,  x  —  ?/  =  di  2  (4) 

Adding  (1)  to  3  times  (2), 

x^  +  2xy  +  ?/2  =  64  (5) 

Extracting  root,  ^  +  «/  =  ±  8  (6) 

Combining  (6)  and  (4),  ic  =  di  5  ;  | 

And  y  =  ±2,,   f 

15.  Given  x  -^  y  =  z']  (i) 
And  xy  =:  1^0  (2) 
Squaring  (i),  ic^  _|_  2xy  -{-  y^  =  729 
Multiplying  (2)  by  4,  4a:*?/  =720 
Subtracting,  x^  —  2xy  -\-  y^  =  g  (3) 
Extracting  root,  ^  —  .V  =  ih  3  (4) 
Bringing  down  (i),         x  -{-  y  z=  2^ 

Combining  (4)  and  (i),  x  =z  1$  or  12 

And  y  ^  12  or  15, 

16.  Given  x  —  y=  14  (i) 
And  xy  =  147  (2) 
Squaring  (i),  x^  —  2xy  -{-  y^  z=  ig6  (3) 
Multiplying  (2)  by  4,          4xy  =  588  (4) 

Add.  (4)  to  (3),    X^  +  2Xy  +  ?/2  r:r    784  (5) 

Extracting  root,  ^  +  i/  =  ±  28  (6) 

Bringing  down  (i),         x  —  y  =1  14 

Combining  (6)  and  (i),  a;  ==  21  or  —  7 ;  ) 

And  ?/  =  7  or  —  21,  ) 

17.  Given  o:^  —  ?/t  —  3  (i) 
And                               a:T  4-  yT  —  7  (2) 

Combining  (i)  and  (2),  ic^  —  5 

And  y^  =  2 
Involving,  ^  =  625  ;  |  ^^^^^^ 

And  y  =  16,      \  ^ 


Aiis. 


SIMULTAKEOUS     QUADRATICS.  159 

1 8.     Given                        ^y"^ -\- x^y-'  =12  (i) 

And                            x^y  -\-  xy'^  =  6  (2) 

Factoring  (i),         xhf  {x  -{-  y)  =  12  (3) 

"         (2),           xy{x-i-y)  =6  (4) 

Dividing  (3)  by  (4),               xy  =  2  (5) 

(4)  by  (5),         x+y  =  3  (6) 

Combining  (5)  and  (6),  x  —  y—  ±1  (7) 
Combining  (6)  and  (7),          re  =  2  or  i;  )     . 

AT  (    Ji-HS, 

And  y  z^  I  ov  2,  ) 

PROBLEMS. 

Page  191. 

1.  Let  x  =  greater, 
And  y  =  less. 

Then  x  —  y  =  4  ( 1 ) 

And  3?  —  y^  =z  448  (2) 

Divid.  (2)  by  (i),     x^-\-xy-\-y^  =  112  (3) 

Squaring  (i),      x^  —  2xy  -\-  y^  =:  16  (4) 

Subtracting  (4)  from  (3),    327/  =  96 

Or  xy  =  32  (5) 

Add.  (3)  and  (5),    x^+2xy^y'^  —  144  (6) 

Extracting  root  of  (6),    x  -\-  y  z=z  ±12  (7) 

Combining  (i)  and  (7),  a:  =  8  or  —  4;  )     , 

And  ?/  =  4  or  —  8,   i 

2.  Let  a;  =  wife's  age, 
Then           a;  +  i  =  man's   '• 
And           ic^  -f  a;  =  930 

By2dmethod,    x  —  —\  ±1^930  + J—— J  +  V^ 
.-.    ir  =  —  |±-%i  =  30  years,  wife's  age;  ) 
And  a;  +  I  =  31      "      man's   "     f 


21 


IGO  SIMULTANEOUS     Q  U  A  D  K  A  T I  C  S  . 

3.  Let      X  =  greater,     and    y  =  less. 

Then  (x  -{-  y)x  —  180  (i) 

And  {x  —  y)y=i6  (2) 

From  (i),  x^  -{-  xy  =z  180 

"      (2),  xy  -7/2=16 

Assuming,  x  =  py 

Substituting,  p^y^-\- py^  =  iSo  (3) 

And  py^  —  y^  =16  (4) 

■^  ,         K  0  180  /  , 

From  (3).  y=^l^,  (5) 

"  (4),  y'^jzi-,         (^) 

■^  .  ,    X  1     /^^  16  180 

Equating  (5)  and  (6),     ^— -^  =  ^^^y^ 

Clearing,  etc.,  \p^  +  4i^  =  45i^  —  45 

Transposing,  ^i9-  —  4ip  =  —  45 

By  3d,    64p^  —  6^6p  -\-  (41)2  =  '—7204-1681  =  961 

Extracting  root,        8;;— 41  =  ±31 

Whence,  8;?  =:  41  ±  31 

Dropping  2d  value,  P  =  9  (?) 

From  (6),  y"^  =  2 

Extracting  root,  y  —  ±^2  ;  |  ^^^^^ 

And  py,     oy    x  —  9 A/2,     \ 

4.  Let  X  =  number  of  rows. 

And  «/  =        "        "  trees  in  each. 

Then  xy  —  1000  (i) 

And  X  —  y  =  15  (2) 

Squaring  (2),     x^  —  2xy  +  ?/2  =  225  (3) 

Multiplying  (i)  by  4,        4^^  :=  4000  (4) 

Adding,  x^  +  2ir?/  +  «/^  =  4225  (5) 

Extracting  root,  .r  +  ?/  =  ±  65  (6) 

Combining  (6)  and  (2),  x  *-  y  —  15  (2) 

X  =  40;    y  =  25,  ^?z.<;. 


SIMULTANEOUS     QUADRATICS.  161 

5.  Let  X  =^  length, 
And                                    y  =  breadth. 

Then                                 xy  =  g6o  (i) 

And                            X  —  y  =  16  (2) 

Multiplying  (i)  by  4,    4xy  =  3840  (3) 

Squaring  (2),  x^—2xy-\-y'^  =  256  (4) 

Add.  (3)  to  (4),  x^  +  2xy  +  f  =  4096  (5 ) 

Extracting  root,         a;  -f  //  ==  ±64  (6) 

Combining  (2)  and  (6),    x  =z  40  yards;  )     . 

>  Ans. 
y  =  24  \ 

This  problem  may  be  solved  by  one  unknown  quantity 

thus : 

Let  X  =  breadth. 

Then  a;  -f-  16  =  length, 

And  x^  -\-  i6x  =  960,  area. 

By  2d  method,  x  =  —  S  ±:  V960  +  64 

Reducing,  x  =:z  —  8^32 

a;  =  24  yds.;    a:  +  16  =  40  yds.,  A718. 

6.  Denote  the  numbers  by  x  and  y. 

Then  x^ -t  y^  —  .(.c  +  y)  =  78  (i) 

And  ^^  +  ^*  +  y  =  39  (2) 

Adding  (i)  to  (2)  x  2, 

x^  +  2xy  -{-  y^  -\-x  +  1/  =  156  (3) 

Or  *(x  +  2/)2  +  (^  +  ^y  =  156  

By  2d  method,  x  -\-  y  —  —  |±  1/156  4-  \ 

Eeducing,  x  +  y  ^=^  —  i  ±  -¥■ 

Dropping  2d  value,  x  -\-  y  ^=:  12  (4) 

Substituting  (4)  in  (2),  xy  =27  (5) 

^  Combining  (4)  squared  and  (5)  x  4, 

x^ — 2xy-\-y'^  =  2>^  (6) 

.Extracting  root,  .9-  —  _?/  =  ^h  6  (7) 

Combining  (4)  and  (7),  x  —  9  or  3  ;  )   ^^^^^ 

And  ,  _?/  =  3  or  9,   f  ^ 

*  Consider  {x  +  y)  as  a  single  quantity.     (Art.  69.) 


163  SIMULTANEOU.S     QUADEATICS. 

7.  Let  X  and  y  denote  the  numbers. 

Then                 x^  +  «/^  +  '^^'  +  ^  =  188  (i) 

And                                          ic?/  =     77  (2) 
Adding  (i)  to  (2)  x  2, 

{x  +  yf  J^  {x  +  y)  —  342  (3) 

By  2d  method,  x  -\-  y  ^  —  -J  ±  a/342  +  J 

Eeducmg,  a;  +  «/  =  18  (4) 

Subt.  (4)  from  (i),  x^  -\-  y'^  =1  i^o  (5) 

Subtracting  (2)  x  2  from  (5), 

x^  —  2xy  4-  ?/2  =  16  (6) 

Extracting  root,  ^— ^=±4  (7) 

Combining  {4)  and  (7),  ;t'  =  11  or  7;   ] 

nd  2/  =     7  or  II,  j 

8.  Let  a;  =  length,  and  y  =  breadth. 

Then  2{x  +  y)  =  100  (i) 

And  xy  ==  $Sg  (2) 

From  (i),  x-^y  =  s^  (3) 

Comb.  (2)  and  (3),  comp.  sq.,  extract,  root,  etc., 

x  —  y  =:  ±  12  (4) 

Combining  (3)  and  (4),  ic  =  31  rods;  )    . 

And  7/  m  19     "      ) 

9.  Let :?:  and  «/  denote  the  numbers. 

Then  xy  =  28  (i) 

And  a;2  +  ?/2  —  65  (2) 

From(i),  2xy  =  s^                    (s) 

Adding,  comp.  sq.,  etc.,  x  -{-  y  —  ^  n               (4) 

Subt.,  comp.  sq.,  etc.,  ^  —  «/  =  ±    3               (5) 

Combining  (4)  and  (5),  x  =  ±7;) 

And  y  =  ±A^  ) 

10.     Let  X  =  number  on  side  of  greater  square. 

And  y  z=       ''  «        '^      less  " 

Then  x^  +  y^  =z  1154  (i) 

And  X  —  y  =       2  (2) 


SIMULTANEOUS     Q  U  A  D  11  A  T  I  0  S  .  163 

Subt.  (2)2  from  (i),  2xy  =1150  (3) 

Add.  (i)  and  (3),  comp.  sq.,  etc., 

x  +  y  =  ±48  (4) 

Combining  (2)  and  (4),  a;  =  25  ;  j  ^^^^ 

And  ^  =  23,   f 

This  problem  may  be  solved  by  one  unknown  quantity: 

Let  cfi  -\-  {x  -\-  2)2  =1154 

Uniting,  comp.  sq.,  etc.,  q:  =  23 ;  ) 

And  a;  +  2  =  25,  j 

11.  Denote  the  numbers  by  a;  and  «/. 

Then                                        ^U  =  zi^  +  y)  (0 

And                                   x^  -\-  i/^  —  160  (2) 
Adding  (i)  x  2,    ar^  -f  2x1/  -\-  y"^  =z  160  -{-  6(x -\- y)  (3) 
Transp.,  etc.,  (x-\-yY — 6{x-{-y)  =  160 
Comp.  sq.,  2d  meth.,  etc.,  .'^  +  ^  ^  3  ±  13 

Dropping  negative  value,  x-\-y  ^  16  (4) 

Substituting  in  (i),                xy  =  4S  (5) 

Combining  (4)  and  (5),    a:  —  «/  =  ±  8  (6) 

(4)     "    (6),  ^  =  12  or4;    /  ^^_ 

And  y  =:    4  or  12,  f 

12.  Denote  the  numbers  by  a- and  7/. 

Then  xy  =z  6{x  —  y)         (i) 

And  a:^  -f  1/2  z=  13  (2) 

Subt.  (i)  X  2,        x^  —  2xy  +  ?/2  =  13  —  12  (:c  —  y) 
Transposing,  etc., 

(x-7jY  +  i2(x-y)  =  13 

Completing  square,  etc.,  x  —  y  ==  —  6  ±  V13  +  36 
Dropping  negative  val.,    x  —  ^  =  i  (3) 

Substituting  in  (i),  xy  =  6  (4) 

Combining  (2)  and  (4)  and  extracting  root, 

•r  +  y=  ±5  (5) 

(3)    "     (5)?  a;  =  3  or  -  2;)^^^^ 

And  ^  =  2  or  —  3,  I 


164 


RATIO. 


3- 

4- 
5- 


14. 
15- 


RATIO. 

Page  195, 

40  I 

40  :  160  —  --—  =  -,  Ans. 
160       4 


1:81=-,  Ans. 

o 

64  :  320  =  —  =  -,  Ans, 

320       5 

6.  8^2  :  4fi^  =  —  z=z  2a,  A71S, 

4a 

J  J        iKabc  , 

7.  isabc  :  sab  —  -^^—  =  3c,  ^ws. 

8.  500  :  50  =  - —  z=  10,  Ans, 
9-     75  :  600  ::.  -/^  =  \ ,  Ans. 


10.     35  :  35  X  4 


2r<^ 


-^^ —  =  - ,  Ans, 

35  X  4        4 


II.        26^2   :    4ff  rr:   ==   - ,    ^?i.<f. 

46«  2 


r'S  7/2 


rz:' 


.2 


12.     X'^  —  y^  :  X  -\-  y  =:  - 


/_. 


X  -\-  y 


X  —  If,  AnSo 


8-15  8        25       4      , 

25  :  30  )  15       30       9 

a  :  b      )  a        2b          2       , 

,  I  =  T  X =  — ,  Ans, 

20  :  $ax  b       ^ax       ^x 


/;         24         I 

17.  24  :  96  =  ^  =  -,  Ans. 

96        4 

18.  144  :  1728  = =  — ,  Ans, 

1728        12 

19.  Ratio  of  equality.     (Art.  355.) 

20.  Ratio  of  equality. 

21.  35  :  7  =  5- 

Hence,  ratio  of  greater  inequality.    (Art.  356.) 


PROPORTION. 


1C5 


22. 

23- 
24. 


6  :  48  =  |.     Ratio  of  less  inequality.     (Art.  357.) 
15  •  9  =  if ;  38  :  19  =  2;  .-.  38  :  19  >  15  :  ij,  Ans. 

8:25  =  2^ ;  V4  :  V^5  =  i,  or  il  .'.  fs  <  If,  Ans, 
Let  ic  =  tlie  consequent, 
56 


Then 


X 


=  S;     8.T  =  56; 


X 


7,  yl^S. 


26.     Let  X  =  the  antecedent, 


X 


Then      -  =  14; 

7 


X  =   98,    ^?^5. 


PROPORTION. 
Page  204. 

2.  Let  :»  =  the  first  term. 
Then                    x  :  ^    ::    6  :  12 
And                  12a;  =  48 

X  =z  4,    ^^.9. 

3.  Let  a;  =  the  third  proportional. 
Then                    25  :  400    : :    400  :  x 

And  25a;  =  160000 

a;  r=  6400,  Ans. 

4.  Let  X  =  the  mean  proportionaL 
Then  g  -.  x    : :    x  i  id 

And  a;2  r=  144 

ic  =  12,  ^4ws. 


Let 
Then 
And 
Equating  (i) 

Reducing, 
And 


X  =  the  greater,  and  y  =  the  less. 


X  :  y    ::    x-\-y  :  42 
X  :  y    ::    i?:;— ?/  :  6 
and  (2),  (Theorem  9), 

x-\-y  :  x.—y    ::    42  :  6 
x-\-y  '.  X — y    ::      7:1 

^  +  y  —  i^  —  iy 


(2) 


Transposing,  etc., 


X 


.  -  ^y 


(3) 


166  PROPORTION. 

Substituting  in  (2),        ~  '  y    '-'-     ~  —y  \  ^ 


o  o 


4  y 

Div.  ist  couplet  by  ?/,      -    :  i    ::     -  :  6  (Th.  6) 

Mult,  antecedents  by  3,     4:1    : :     ^  •  6 

y  =^  24,  less:  I  ^^^^^ 
Substituting  in  (3),  x  =  :^2,  gr.,    f 

6.  Let  X  =  one  part,  and  18  —  a;  =  the  other. 
Then  x^  :  (18— a;)^    ::    25  :  16  (Th.  12) 
Extracting  root,        a;  :  18 — x    ::    5:4 
Eeducing,  4X  =  90  —  5^ 

r?;  z=  10;  ) 
And  i8-a:=    8,  \^'''' 

7.  Let             ic  =  the  greater,  and  28  —  x  =  the  less. 
Then  -— :  ::    32  :  18 

28  —  it'  X 

Reducing,  gx^  =  16  (28  —  xY 

Extracting  root,  (Th.  12),      32;  =  4  (28  —  x) 
Transposing,  jx  =  112 


And 


X  =  16:)    . 

28  —  ^   =    12,     J 


8.     Denote  the  numbers  by  x  and  y. 

Then  xy  =  24  (i) 

And  x^—y^  :  {x—yY    ::    19  :  i  (2) 

Dividing  hj  x  —  y, 

x^-\-xy-{-y^  :  x^ — 2xy  +  y^    ::    19  :  i 
By  Theorem  8,    ^xy  :  {x—yY   ::    18  :  i 

"  "        6,      xy  :  (x—yY    ::      6:1 

"  "        I,  6  (x  —  yY  =  ocy 

Subst.  value  of  xy,      6  (x  —  yY  =  24 
Eeducing,  x  —  y  =^    2  (3) 

Combining  (i)  and  (3),    a;  +  ?/  =  10  (4) 


a 


(3)  "  (4),     -  =  oa^^, 

=  4>  ) 


And  y 


I>t?OPORTtON-.  16'^ 

Denote  the  numbers  by  x  and  y. 
Then  x-\-y  \  x — y    ::    9  :  6  (i) 

And  '^^—y  '  xy    ::    1  :  12  (2) 

Reducing  (i)  by  Theorem  (6), 

x-\-y  :  X — y    1:3:2 
By  Theorem  7,  2X  :  x — y    : :    5:2 

I,  4^  =  5  {^  -  y)       (3) 

Eeducing  (2)  by  Th.  i,  xy  =  12  (x  —  y)       (4) 

Dividing  (3)  by  (4),  etc.,        sy  =  48 


iz;  =  48,     f 


ri  ns. 


Substituting  in  (3), 

10.  Let  X  =  the  length  in  rods, 
And  y  =    "    breadth       " 
Then  the  area,  xy  =  860  x  160         (i) 
And  X  :  y    ::    4S  :  32            (2) 
Reducing  (2),  322;  =  43?/ 

And  x  =  ^^  (3) 

32  ^"^^ 

4"??/^ 
Substituting  in  (i),  ^^^^-  —  860  x  160 

o 

T^    1       .  „  8S0  X    160  X  32 

Reducino-  ?/^  = -^^ 

""  -^  43 

Cancelling  43,  y'^  =  6400  x  16 

Extracting  root,       ?/  =  80  x  4  =  320  r.  breadth; 
Substituting  in  (3),  x  =  430  r.  length. 

1 1.  Let  X  =  side  of  one, 

Then  10  +  .'c  =     "     "  the  other. 

And  x^  :  (x  i-  lo)'-^    : :    4:9 

By  Theorem  12,     ic:a;+io    ::    2:3 

"  '*'  I,  3:c  ==  2:?;  +  20 

a;  =  20  rods ; )    , 
*     -.  ..    '  \  A71S. 

And  10  +  a;  =  30     "      ) 


1G8  ARITHMETICAL     PROGRESSION". 

12.     Denote  the  numbers  by  x  and  y. 

Then  xy  =  135  (i) 


And  x^  —  y^  :  {x  —  yf 

By  Theorem  6,   x  -\-  y  :  x  —  y 

"  "        7,         2T  :  2;  —  y 


4  :  I 

4  :  I 

5  '  I 


(6 


«  "        I, .  2a;  =  5^  —  5?/ 

And  X  =  -^  (2) 

3 

Substituting  in  (i),  ^^  =r  135 

Reducing,  2/^  =  81 

Extracting  root,  y  =  g\  \ 

Substituting  in  (2),  2;  =  15,  i 

13.     Denote  the  numbers  by  x  and  y. 

Then  ^?/  =  320  (i) 

And  x^—  y^  :  (x  —  yY    : :    61:1 

By  Theorem.  6,  x^-\-xy-\-y^  :  x^ — 2xy-\-y'^    ::   61  :  i 
"        2>,  ^xy  :  x^  —  2xy  -\-  y^    : :    60  :  i 

"         I,  ^xy  ^  60  (x^—2xy-{-y'^) 

Dividing  by  3.  20  {x  —  yY  =  xy 

Subst.  val.  of  xy,        20  (x  —  yY  =  ^20 
Dividing  by  20,  (2;  —  ^)^  =  16 

Extracting  root,  x  —  y  =z  4  (2) 

Combining  (i)  and  (2),     x  -{-  y  =  s^  (3) 

"  (2)    "     (3),  .-r  =  20 ;  )    . 

And  y  =  ^^,  ) 


ARITHMETICAL    PROGRESSION. 

Page  207- 

2.  /  =  «  ±  (^  —  i)  ^  =  25  +  (9  —  1)  (—  2) 

=3  25  —  16  =  9,  Ans, 

3.  ?=  12  +  14  X  4  =  68,  .1^5. 

4.  ?  =  I  —  12X2  =  —  5'  ^^^^* 

5.  Z  r=  J  -f  8  X  -J  =  ij,  ^1/iS. 


ARITHMETICAL     PROGRESSION.  169 

6.      I  =^  I  —  9  X  .01   =  .91,  Ahs. 

8.  ^  :^  I  +  14  X  3  =  43?  ^n^' 

9.  Z  =  31  —  8  X  2  =  15,  A71S. 

10.  Z  =  I  +  29  X  i^  =  44h  ^*^^- 

11.  Z  =  a;  +  24  X  2a;  =  4gx,  Ans. 

12.  Z  =  2a  H-  (/i  —  i)  3a  =  3«7i  —  a,  Ans, 

Page  208. 

2.  s  ■=  X  /i ;  therefore, 

2 

s  =  ^ ^  X  50  =  762J,  JW5. 

3.  s  =  -^  X  9  =  216,  Ans, 

4.  s  =  ^  -  X  35  =  1400?  ^^i5« 

2 

5.  5  =  — ' —  X  17  =  25i,  A)is, 

2 

6.  ?  =  2  +  19  X    3  =  59 ;  and 

s  ■=.  X  20  =  610,  Ans. 

2 

7.  /  =  I  +  24  X    i  =  13 ;  and 

s  = X  25  =  175,  Ans. 

2 


8.     Z  =  75  —  14  X   3  =  33  ;  and 
75  +  33  ^  15  =  810,  ^?2s. 


2 


1.  ?  =  25  +  II  X  3  =  58,  Ans. 

2.  ?  =  58  +  44  X  5  =  278,  yl?k<?. 

^     fi{  r=  Z  —  (m  —  i)  ^7  =  35  —  8  X  3  =  II,  Jw5. 
8 


170 


AEITHMETICAL     PROORESSIOK, 


4. 

a 

5- 

d 

6. 

d 

7. 

n 

8. 

n 

57  —  20x5 


43,  Ans. 


l-,a  _  85-15  __    .     .„5 

71—1  30 

7  —  28  21       . 

= ,    A71S, 

25 


25 


1152  —  6 


c;n8  —  2^  . 

+  I  ==  ^-^ ~  4-  I  =  1024,  A71S, 


-\-  I  =  T92,  Ans, 


I.     s 


2.     £ 


X  01  = 

2  2 


rage  210. 

9  +  41 


X  7  =  175,  Ans. 


+  45 


X  50  =  1 130,  Ans, 


3- 

« 

4. 

^ 

5- 

a 

6. 

n 

7. 

I 

8. 

I 

9- 

I 

I  —  (n—  i)d  =  50—11x4  =  6,  A71S. 

2S  2    X    150  ,        . 

a  -\-l        9  +  41 

/  +  (^  —  i)  (?  =  21  +  34  X  7  =  259,  Ans. 

2S     _  2  X  455 
a  -\-  I        46  +  24 


=  13,  Ans. 


2s              2  X  72  . 
a  =i 27  =  —  II,  Aiu 

9 


n 


2  X  288  . 

z=  ■ 72   z=  O,   A71S, 

=  3  +  14x2  —  31;  and 


s  = 


3  +  31 


X  15  =  255,  Ans. 


10.     Z  =  5  +  19  X  3  =  62,  A)is. 
XI.     t  r=  5  +  14x4  =  61,  Ans. 


AilltHMETlCAL     PROCRESSIOK".  itl 

FORMULAS. 

Page  211, 

9.     Given  d,  n,  and  s,  to  find  a,  tlie  first  term. 

By  Formula  (2),  s  — x  n 

Siibst.  for  I  its  value  m  (i),    s  = > -'—  x  n 

2 

2  e  ..^_  fjiy^     I     fl'}'). 

Multiplying  by  2,  etc.,  a  =  — • 

10.     Given  d,  I,  and  s,  to  find  a,  the  first  term. 
Equating  Formulas  (5)  and  (8), 

I  —  a  _     2s 

~d~  "^  ^  ~  «+l 
Clearing  of  fractions, 

12  _  ^^2  _|_  ^f/  -I-  f^/  =   2^.§ 

Transposing,      a^  —  da  ^  1^  +  dl  —  2ds 


d           I              d? 
Completing  sq.,  etc.,    «  =  -  ±  A  /  'l?-\-dl-\ 2ds 


Changing  the  form,     «  =  -  ±  A  /  (Z  +  - ) 


2ds 


11.  Given  a,  1,  and  .<?,  to  find  d,  the  common  difference. 
Transposing  in  Formula  (10),  and  squaring, 

a^  —  ad  =^  P  -{-  dl—  2ds 
Transposing,       2ds  —  dl  —  ad  =:  P  —  a^ 

Factoring,  etc.,  d  = , 

12.  Given  Z,  n,  and  s,  to  find  d,  the  common  difference. 

2  Q dfi  -A-  dfi 

Equating  (3)  and  (9),   l—(n  —  i)d=  2n~~~ 

Mult,  by  2n,         2ln  —  2dn^  +  2dn  =  2s—dn^  +  dn 
Transposing,  dn^  —  dn  =  2ln  —  2s 

Factoring  and  dividing,  d  =   -^ y  • 


172  ARITHMETICAL     PRO  GKESSIOK. 

13.     Given  a,  n,  and  ^,  to  find  d,  the  common  difference. 

2* (I'li^  -4-  (111 

Bv  Formula  (9),  a  = 

Multiplying  by  2/?,  2an  =  25  —  6?^^  -f  dn 

2S  ^—  2(171 

Transposing  and  dividing,       d  = 


n^  —  n 


14.  Given  d,  n,  and  s,  to  find  ?,  the  last  term. 

By  Formula  (12),  d  =  ^-^7 — ^^ 

•^  ^     ^  n(n—  i) 

Eemoving  denom.,     w(/^ — i)d=:2hi  —  2s 
Transposing  and  dividing,       I  =  — f-  ^^ —  • 

jv  2 

15.  Given  a,  d,  and  s,  to  find  I,  the  last  term. 

^ ojt 

By  Formula  (11),    d  := , 

Kemoving  denominator, 

2ds  —  dl  —  ad  ^  '^  —  €? 
Transposing,     '^-\-dl  =^  zds  —  ad  •{-  a^ 

d  I  ^ 

Comp.  sq.,  etc.,         I  •=. ±  \     2ds  +  a^—ad-\-  — 

d          I           /        ^ 
Or,  ?  = ±  A/  2^s  +  I « 1  • 

16.  Given  a,  d,  and  s,  to  find  n,  the  number  of  terms. 
By  Formula  (13),  d  =  ~^— 

Removing  denom.,  etc., 
dv?  -{-  {2a  —  d)  n  =  2S 

Completing  square,  etc., 

2d7i  -{-2a  —  d=  ±  \/(2a  —  df  +  Ms 

Transposing  and  dividing, 

±^/i2a—df-{-Ms—2a-\-d 
2d 


ARITHMETICAL     PROGRESSION.  173 

17.  Given  d,  I,  and  s,  to  find  n,  the  number  of  terms. 

By  Formula  (12),      d  =  ^, — "^—l 
•^  ^     ^  n(7i  —  1) 

Removiug  denominator, 

dn^  —  dn  =  2ln  —  2  s 

Transposing  and  factoring, 

^  2I  4-  d  2S 

n^ -, —  n  =■ 7 

a  d 

Completing  sq.,  etc.,  n  =  — ^  ±  \/ ^-^ 

^        ,     .                          2l  +  d±  V(2T^~dy^Sds 
Or,  reducmg,  71  =z —^ 

2Clr 

18.  Given  a,  d,  and  n,  to  find  s,  the  sum  of  the  terms. 

2S  —  dii^  +  dn 
By  Formula  (9),  a  = 

271 

Multiplying  by  2n,  2an  =  25  —  dv?  +  d7i 

Transposing,  2s  =  2  an  +  dn^  —  dn 

s  = -[2a-{-{n  —  i)d]. 


2S 

~d 


2 


19.  Given  a,  d,  and  I,  to  find  s,  the  sum  of  the  terms. 

By  Formula  (11),  <?  = ^ 

Remov.  denom.,       2ds—dl — ad  =  P  —  a^ 
Transposing  and  factoring,  2ds  =  (l-\-a)d  -\-  P—  a 

Dividmg  by  2d,  -     _       s  = 1 -j — 

20.  Given  d,  I,  and  n,  to  find  s,  the  sum  of  the  terms. 
By  Formula  (14),  Z  =  -  +  ^^ ^— 

Transposing,  -  =  I  —  - — — 

"     Multiplying  hy  n,  s  =  -  [2I  —  {n  —  1)  d] 


2 


174  ARITHMETICAL     PROGRESSION^. 

Page  212, 

1.  d  := =  6.    Hence  the  series: 

5 

I,  7,  i3j  19?  25,  31,  A71S. 
.0  ___  _ 

2.  d  ■= =  4I.     Hence  the  series: 

10  ^ 

3,  71,  12,  i6i   21,  25I,  30,  34I,  39,  43^,  48,  A71S. 


PROBLEMS. 

I.  /  =  5  +  14  X  3  =  47?  Ans. 

2.  Z=:27  —  11X3=—  6,    ^^5. 

3.  l=z'j-^igK$=^   102,   yl?iS. 
,/  — ^60  —  2  - 

^  /??  +  I  6  ^^ 

Hence  the  series:  2,  iif,  21},  31,  4o|,  50J,  60,  Ans. 

5.     /  z=  «  +  (;^  —  i)  r?  =  i  +  99  X  i  =  33  J ;  and 

5  =:  — ^—  X  ?i  =  l^Jl-^^  X    100  =:    16834,    ^^'^'• 
2  2  ^"^ 

-      ,        2s—2a7i        2x18750  —  2x5x20         _,      . 

W^  —  71  400  —  20  ^ 

7.  ?  r=  I  +  75  x  2  =:  151  ;  and 

1  +  151  ,  .A 

s  = — ^^  X  76  r=  5776,  Ans. 

8.  Z=2  +  99X2  =  200; 

2  +  200  . 

s  = X  100  =  loioo,  Ans. 

2 

rage  213. 

-,        I  —  a       47  —  2  . 

9.  d  =2 =  — =  5,  Ans. 

^  n  —  I  9  ^ 

,        I  —  a         72  —  6 

10.     f/  = = =  il. 

in  +19'^ 

Hence  tlie  series: 

(>,   I3i^  20f,  2%^  35J,  42|,    50,    57-J,   64J,   72,    J725. 


AKITHMETICAL     PliOGRESSlON".  175 

/  —  a  1 08  —  12 

11.  d  =. =r  —  9.6 

w  4-  I  10  ^ 

Hence  the  series: 

12,  2  1.6,  31.2,  40.8,  50.4,  60,  69.6,   79.2,  88.8,  98. j|, 

108,  Aiii^. 

12.  I  =  100  —  14  X  5  =  30;  and 

100  +  "^O  . 

^  — ^  X  15  =  975.  ^'^5- 

14.  Let  X  =  the  second  nnniber, 
And                              y  =  the  common  difference. 
Then,  x^y  +  x-{-x-{-y  =  15;  or  32:=  15,  a;  =5.   (i) 

And  {x—yy  +  x^+{x  +  yY  =  49S         (2) 

Expanding  and  reducing,     30.^4-62:^'^  =  495 
Substituting  value  of  .r,  etc.,         30?/^  =120 

y  =  ±2 

Hence  the  numbers:  3,  5,  and  7,  Aiis, 

15.  Since  he  lias  to  pass  over  the  ground  twice  for  each 
marble,  the  problem  requires  us  to  find  twice  the  sum  of 
the  series. 

By  Formula  (i),     I  =  a  -i-  (n  —  1)  d 

I  z=    I   -f  99   r=    100 

By  Formula  (2),    s  = x  7i 

2 (i + 100) 100  , 

25  =  -^ =  10 100  yds. 

==  51  miles  nearly,  Aus. 
Page  214. 

16.  By  Formula  (2),  2s  = —  x  12  =  156,  Ans. 

17.  By  Formula  (i),     7=  10  +  24  x  20  =  I4.90 

«        «         (2),    s  =  ;^-  X  25 

8  =  $2.50  X  25  =  $62.50,  Ans, 


176  ARITHMETICAL     PROGRESSION. 


1 8.  By  Formula  (2),    s  =  — "tJ-l  x  365; 

s  =  183  X  365  =  I667.95,  ^^^« 

T      „1„     *^  A 

19.  By  Formula  (2),    s  = x  24  =  300,  Ans. 

20.  By  Formula  (i),    ^  =:  .06  +  19  x  .06  =  I1.20,  int. ; 

li.oo  +  I1.20  =  $2.20,  amount,  Ans, 

21.  Denote  the  numbers  by 

a  —  d 

r 

a 
And  a  -^  d 


Then  30^  =  120;     a  =  40, 

And  3^2  +  2d^  =  5600 

d  ■=.  20 

Hence,  a — d  =z2o,a  =  40,  and  a  +  d  =  60,  ^ws. 

22.     Let  n  =  No.  of  days  the  2d  travels, 

Then         30  +  ion  =    "    "  miles "    ist      " 
By  Formula  (i),      I  =  4  -\-  {n  —  i) 

/  =  3  +  ?^ 

By  Formula  (2),     s  =  - — xn  =  dis.  2d  goes. 

Equatnig,   -5^--^^—  =  30  +  ion 

Mult,  by  2,     fn-j-n^  =  60  +  2on 
Transpos.,  n^—i$n  =  60 


Comp.  sq.,  etc.,      n  =  ^  ±  \/6o  +  -3^ 

71  =:  16.61+  days,  A71S. 

23.     Denote  the  numbers  by 

ff  —  3^?,  a  —  d,  a  -\-  d,  and  a  +  3d, 
Square  of  ist,  a^  —3f^(^  +  9^^ 

"       "  4th,         r?^  4- 6flfr/  4-  9^<^^ 

'      Sum,  2^2  +  i8(?2  =  4500  (i) 


ARITHMETICAL     PROGRESSION".  177 

Square  of  2d,  a^  —  2 ad  +    d^ 

«       "  3d,  a^  -{-  2ad+    ^ 

Sum,  20?  +  2f/2  =  4100  (2) 

Subtracting  (2)  from  (i),         i6d^  =  400 

d  :=  ^ 
Substituting  in  (2),  2^2  —  4050 

«^  =  45 
Hence,         cj  —  3^/  =  30,    a  —  d  =  ^o,  ) 

«  -}-  ^  =  50,  «  +  3fZ  =  60,  j  ^ 

24.  Let  ?z  =  No.  of  days  B  travels. 
By  Formula  (i),     I  =  7  -}-  {71  —  1)  2 

=  distance  A  goes  the  last  day. 

?  =   5   +   2W 

By  Formula  (2),     s  =  ^     —  x  n  =  6n  +  n^, 

2 

Hence,     g -\- 6n -{- 71^  =  No.  of  miles  A  travels. 

By  Formula  (i),     I  =  11  +  71  —  i  =  10  -{-  n 

=  distance  B  goes  the  last  day. 

■D    17         1    /\  II^-Io  +  ?^  21??  4- 7^2 

By  Formula  (2),     5  = ' ■ —  x  n  =  — 

2  2 

=  B's  distance. 

Whence,  g  +  671  +  71^  = 

2 

Reducing,  w  =  3  or  6  days,  A7is, 

25.  By  Formula  (i),    l=io  —  2oxi  =  ^ 

10  -I-  -ip- 

"        "         (2),    5  = ■ — ?-  X  21  =  140,  Ans. 

2 

26.  By  Formula  (i), 

?  =  I  +  59  X  3  =  $178  last  payment;  ) 

By  For.  (2),   .9  =  i±-il5  x  60  =  $5370  debt,  (  ^^^' 


178  GEOMETRICAL     PKO  GRESiSI  0  K. 

GEOMETRICAL    PROGRESSION. 

rage  210. 

1.  I  ;=.  af'~^  =  5  X  2^  =:  i6o,  Ans, 

2.  Z  =  2  X  3"^  =  4374?  ^ns, 

3.  Z  =  72  X  (4)4  =  4i  ^^^s. 

4.  Z  =  5  X  4^  =  320,  Ans, 

5.  ^  =  7  X  2^  =  112,  Ans, 

6.  ^  =  10  X  (—  5)^  =  —  31250,  A71S, 


rage  217- 

Ir  —  a        2000  X  5  —  8  „      . 

2.  s  = = — =  2498,  Ans, 

r  —  1  4 

5000  X  10  —  9  , 

4.     5  =  £_5QOQ  ^  "^  ~  ^  =  3333 1 1?  ^^«- 

3 

20  X  6  —  15  . 

c.      s  = =  21,  ^?i5. 

^  5 

12  X  4  —  25         ,     . 

3 

rage  218, 

I.     ?  :=  «r"~^  =  3  X  10*  =  30000,  Ans» 

2.        ?  =  5    X   5^  =   15625,   ^W5. 

/         256  . 

3.  «  =  ^1  =  -^  =  2,  ^^is. 

4.  «  =  -^  =  3,  ^;^s. 

5-     "  =  (it  =  iP?f  =  ^'  ^- 

6.     r  =  f-^ — y  =  1/25  X  25  =  5,  Anci, 


GEOMETEICAL     PROGKESSION.  179 


I.       S  = 


Page  21i). 

Ir  —  a        io8  X  3  —  2 


r=    l6l,    A71S. 


r  —  I  2 

2.     «  =  /r  —  -s  (r  —  i)  =  54  X  3  —  8o  X  2  =  2,  A7is. 

^'                      r  5 

s  —  a  15624  —  4                  . 

^              s  —  I  15624 — 12500 

Ir  —  a  150  X  6  —  5                  .    ^ 

5.     6'  =  -^^---  =  -'-- ^  =  179.  ^4^.5. 

r  10 


FORMULAS. 

Given  ??,  r,  and  *',  to  find  a. 
By  Formula  (i),  I  =  ccr"-'^ 

Multiplying  by  r,  Ir  =  af' 

Substituting  in  For.  (6),  a  =  ar"  —  (r  —  i)  s 

Transposing,  etc.,  «r"  —  a=  {r  —  i) s 

{r-i)s 


/•"  —  I 


Factoring,  etc.,  a  = 

10.  Given  I,  n,  and  s,  to  find  a. 
Equating  Formulas  (4)  and  (8), 

sj-a  _  //\„4r 
s  —  I        \al 
'Invohnng  and  clearing  of  fractions, 
a{s  —  aY-'  =  lis  —  iy-K 

11.  Given  «,  71,  and  s,  to  find  I. 

Transp.  For.  (10),      I  (s  —  I}"-^  =  a  (^  —  ay-K 


180  GEOMETRICAL     PKOG  R  ESS  lOl^". 

12.  Given  n,  r,  and  s,  to  find  I. 

Equating  Formulas  (3)  and  (9), 

I    _  (r  —  i)j 

a»n— 1  /pll    T 

/^»  i\  g^n—\ 

Multiplying  by  r-\      I  =  ^^-^-"1 

13.  Given  a,  I,  and  s,  to  find  n. 

By  Formula  (10), 

a{s  —  «)«-'  =  l{s  —  0»-i 
By  logarithms. 

log.a-^\og.{s—c(){u—i)  =  log./  +  log.  (s—l)(n  —  i) 
Transposing, 
log.  {s—a)(7i—i)—\og.  (s—l){7i  —  i)  =  \og.l—\og.a 

Factoring,  etc., 

log.  I  —  log.  a 


71  —  I  = 


log.  {s  —  a)  —  log.  {s  —  I) 


r^  .  log.  /  —  log.  a  , 

Transposing,     «  =  to^,-Tr^^T^g.-(J^7)  +  '' 


14.     Given  a,  r,  and  s,  to  find  71. 

By  Formula  (9),  a  =  —r^- 

Kemoving  denominator, 

ar"  —  a  =  (r  —  i)s 
Transposing,      at"  =  «+(/•  —  1)5 
By  logarithms, 
log.  a  +  log.  r  X  /^  =  log.  [rt  -\-  (r  —  i)  s] 

Transp.,  etc.,        7i  =  ^~^- YogT^"^ ^ — 


GEOMETllICAL     f  ROG  R  ESS  J  0  ]^.  181 

15.  Given  I,  r,  and  s,  to  find  n. 

By  Formula  (12),  I  =       ^,  _  ^ 

Removing  denominator,        Ir""  —  I  =  (r  —  i)  sr"^~^ 
Transposing,      In  —  (r  —  1)  sr"~^  =  I 
Factoring,         [Ir  —  (r  —  i)s]r"'^  =  I 

Dividing,  r"~^  =  j . r- 

°'  Ir  —  (r—i)s 

By  logarithms, 

log.  r  X  {n—  i)  =  log.  I  —  log.  [/r  —  (r  —  i)  s] 

Dividing  and  transposing, 

_  log.  /  -  log.  [Ir  -  {r  -  1)5] 

log.  r 

16.  Given  a,  w,  and  5,  to  find  r. 

By  Formula  (9),  a  =  ~^P~^ 

Removing  denom,,         «r"  —  a  =  rs  —  s 

Transposing,  ar"^  —  rs  ■=  a  —  s 

s  s 

Dividing,  y" r  =  1  —  -• 

17.  Given  /,  n,  and  s,  to  find  n 

^^' j\  ^/•"■~* 

By  Formula  (12),  1=       ,,n  _  j  — 

Removing  denom.,  Irn  —  I  =z  (r  —  i)  sr"~^ 

Transposing,  Zr"  —  (r — i)  sr"~^  =  / 
Or,  Ir  —  sr"  +  sr"-^  =  I 

s  I 

Factoring,  etc.,    r"  -f-  j r""^  =  ,  __-» 

18.  Given  a,  n,  and  r,  to  find  s. 

By  Formula  (9),  «  =  -  ^  _  — 

Removing  denom.,     a  (r«  —  i)  =  {r  —  i)  s 

^.  .,.                                                 «(r''-  I) 
Dividmg,  s  = -_ • 


182  GEOMETRICAL     PROGEESSION. 

19.     Given  /,  Uy  and  r,  to  find  s. 

By  Formula  (12),  I  =  fr~'^^'^-- 

r"  —  I 

Removing  denom.,  Ir'^  —  I  =  (r"  —  r"~^)  s 

Ir"  —  I 


Dividinsf 


s  = 


20.     Given  cj,  Z,  and  ^i,  to  find  s. 

By  Formula  (4),  r  =  ^ 

— T^  —  a 

/a 
Substituting  in  For.  (2),  5  =r  —-^ 

n — 1/ 

Multiplying  both  terms  by  '^V^? 

_    v/"  — ^  V«" 
v?  —  v« 


2,       T  —  f-V"^^   =   (128  -^  i)^  =   V^6   =r  4. 


^?^5.  \,  2,  8,  32,  128. 


I.     By  Formula  (2),    5  = 
Substituting, 


PROBLEMS. 
Ir  —a 


r  —  I 

2916x3  —  6  . 

=  ~ =  437 1 J  A71S, 


2,     By  Formula  (18),  5  =  ,  or  s  = 

Substituting,  s  =  ?-— :i:Aill_  =,  2 rrm 

16  40         .(.,.. 


GEOMETRICAL     PKOGKESSION.  183 

By  Fomiulu  (i),     I  —  ar""'  =  i  x  3^^; 

^  '^             r  —  I             2  2 

(243)^  —  I  . 

=  ^^^^ =  7174453.  ^^^•^'• 

2 


2il2 


4.  By  Formula  ( I ),    I  =  i  x  (f)"; 

«         "        (2),    s  = =  '-^ — I— 

^  ^  r  —  I  —  ^ 

—    ,17  3  0  5  1         A  )iq 

5.  By  Formula  (i),  /  =  nf'-^  —  2  x  3^^—9565938,  Ans. 

6.  By  Formula  (i),   /  =  3  x  3^^  =  43046721,  Ans, 


rage  222, 

8.  By  Formula  (i),    /  =  «r"~^ 

/  =  I  X  2^1  =  2048 

By  Formula  (2),    s  = 

2048  X  2  —  I        .  . 

s  =  — ^^— =  '^4095?  A?is. 

9.  By  For.  (i),     Z  =  i  x  3^  =  19683, 

«      «    /2^      „^  19683x3 -I  ^59048 
^  ^'  3  —  1  2 

=  I295.24,  entire  cost; ) 
$196.83,  last  cow,       \ 

10.  By  Formula  (i),    ?  =  i  x  2^  =  512 

(2),    s  =  5iiJX_l^I  =:  $10.23,  ^l;i5. 
2  —  I 

11.  By  the  conditions,    a-^ar  +  ar^  =:  26  (i) 

And  «2  ^  ^^2  ^  cf2)A  —  364  (2) 

Transposing  ar  in  (i)  and  squaring, 

a^  H-  2rtV2  +  rt'H  =  676  — 52ar  +  «2r2 
Reducing,        a^  +    a^r^  +  a^/"^  =  676— 52«r       (3) 


184  GEOMETKICAL     PKOGRESSION. 

Eq.  (2)  and  (3),         676— 52fljr  =  364 

ar  =  6  (4) 

^  =  r  (5) 

Substituting  in  ( I ),  -  +  6  -{-  6r  =  26 

Reducing,  r^  —  ^r  =  —  1 

Completing  square,  etc.,  ?'  =  3 

From  (5),  a  =  2 

Hence,   a  =  2,     ar  =  6,    ar^  =  18,  Ans, 


31 


12.  By  Formula  (i),     I  =z  i  x  2^  =  2 

(2),       S  =   ^''   ^   ^  ~  '    :=    232  _  I 
^  2  —  I 

=z  $4294967.295,  Ans. 

13.  By  the  conditions,  a  -\-  ar  -\-  ar'^  =  130 
And  ar  -j-  ar^  +  ar^  =  390 
Factoring,  «(i  +  r  +  r^)  =  130 

"  «r  (i  +  r  +  r^)  =  390 

Dividing,  ^'  =  3 

Substituting,  ^(1  +  3+9)  =  130 

«  =  10,    057*  =  30,    «r2  =  90,    ar^  =  270,  ^^6\ 


rage  223, 

14.     By  the  conditions,    a -\-ar-\-ar'^  =1  210  (i) 

And  « —  ar^  =z  90  (2) 

^'•"'n  (')'  « =  t:^^:^      (3) 

Equating  (3)  and  (4),  — 


Reducing,  7  —  7/*^  =  3  H-  3?'  +  3^^ 

Completing  square,  etc.,  ^*  =  | 

Substituting  in  (4),  a  =  120 

Hence,     a  =  $120,    ar  ==  $60,    ar^  =  $30,  ^W5. 


GEOMETRICAL     PROGRESSION.  185 

15.     By  the  conditions,      a-\-ar-}-ar^  +  ar^  =  46S       (i) 
And  ar  +  ar^  +  ar^  +  ar*  =  2340     (2) 

Factoring,  a (i  -{-  r  -\-  r^  -\-  7-^)  =  468       (3) 

«  ar  (i  +  r  +  r^  +  r^)  =:  2340 

Dividing,  ^  =  5 

Substituting  in  (3),  1560!  =  468 

a  =  3 
Hence  the  numbers  are  3,  15,  75,  375,  1875,  Ans. 


16.     Denote  the  shares  by  — ,  cc,  y,  and  —  • 

y  a; 

Then  —  +  '^'  +  2/  +  "|-  =  7oo  (i) 

And  ^^^ :  y—x    ::    37  :  12  (2) 

Clearing  (i), 

Adding,         2X^y  -\-  2xy^  =  2X-y  -f  2^^^         (Ax.  2) 
-  (2:  +  yY  =:  -jooxy  +  2iry  {x  -\-y)    (3) 

Reducing  (2),      ^—  :  ^/-a:    ::    37  :  12 


Dividing  by  {y  —  x),  (Theorem  6), 
y^  +  xy  +  x^ 


37  :  12 


xy 

Multiplying  by  xy,  (Theorem  6), 

y^  -\-  xy  -{-  x^  :  xy    ::    37  :  12 
Or,  (Th.  7),  (y  +  x^  :  xy    ::    49  :  12  (4) 

Multiplying  hj  y  -\-  x,  (Theorem  6), 

(y  +  :r)3  :  xy    ::    4g{y  +  x)  :  12 

Or,  (,  +  ,)s  =  49^(£±1)     (5) 

Equating  (3)  and  (5), 

,        4QXU  (x  +  y) 
•jooxy  +  2:^7/  (.r  +  y)  =  '^^—-^ 

Reducing,      8400  +  24  {x  +  ?/)  =  49  {x  4-  y) 

And  ^  +  y  =  33*^  (6^ 


186  GEOMETRICAL     PROGRESSION. 

Substituting  in  (4),  ^^xy  =z  (Z2>^Y  x  1 2 

xy  =z  (48)2  X  12 
Multiplying  by  4,  /^y  =  (48)3 

=  110592  (7) 

Squaring  (6),  (x  +  yY  =  (ss^f 

=  112896  (8) 

Subt.  (7)  from  (8),        {y  —  xY  =  2304 
Extracting  root,  y  —  a;  =  48  (9) 

Combining  (6)  and  (9),    y  -\-  x  ^  336 

y  =  I192; 

x  =  I144; 

—  :=  I108;  /   A71S. 

y 

^  =  I256, 

* 

17.     By  Formula  (i),  ar"~^  =z  I 

"Substituting,  loooor''  —  1464. 

Extracting  square  root,      loor^  =121 

r  =  fj,  or  I.I,  A71S, 


«  "        "  lor  =  II 


^  ?/ 

18.     Denote  the  numbers  by  ~,  x,  y,  and  —  • 

'/  X 


J*2 

Then         + 

y 

And  -^4-  X 

y2 

'-\-y'  +  \,     85 

Assume 

2^4-  i/  _  5 

a 

^2/  -  ^ 

Then 

.T^  4-  v'^       '^^  —  "^p 

And 

^    _|_    yZ                 ^3    ^^p 

From  (i), 

X^          if' 

•     ^''      (^)^ 

^+J  =  «5     ^+V^ 

(0 

(2) 


(3) 

(4) 


INFINITE     SERIES.  18T 

Squaring  (3),     --^  +  -^  =  225— 30.S  +  6-2-2;;      (5) 

Equating  (4)  and  (5), 

85  —  5^  +  2])  =  225  —  305  +  5^  —  2p 
Transposing,      85  +  47;  =  225  —  305  +  26^        (6) 

From  (3),  Q^  +  y^  =  ^5P  —  sp 

Or,  s^  —  2,sp  =  ISP  —  sp 


Substituting  in  (6), 

85   +  ^7^—    =   225  -  305  +  2S'2 

Removing  denominator,  etc., 

35*2  -j-  ijs  =  210 
Comp.  sq.,  etc.,  s  =  6    or   ct-\-i/  =  6        (8) 

Subistituting  in  (7),       ;;  =  8    or        xy  =z  S        (9) 

Comb.  (8)  and  (9),  x—y  =2  (10) 

«      (8)    "  (10),        x  =  4.    y  =  2, 

x^  ifi 

And  —  =  8,  and  -^  =  i.     Hence,  8,  4,  2,  i,  Ans. 

y  ^ 


INFINITE     SERIES. 
Pnge  227, 

4.     I  —  X  )  I  H-  .1'  (  I  +  2a;  +  2x^  -\-  2x^  H-  2x^,  etc.,  Ans. 
I  —  .r 


2X 

2X  — 

2X'^ 

2X^ 

27?'  — 

2.1*3 

2;7'3 
20^3  _ 

zx^,  etc. 


188 


INFI]SriTE     SERIES. 


rage  228. 

x^  —  y^\  if       y^         if       ,        . 


2X  —  ^- 
2X 


—  i/2 
-f+^ 


20;  —  -^ 

X 


8a^ 


4X^ 

—  ^, + 


f 


+ 


y^ 


4^^       Sic^       64^;^ 


2x^t^y^^  '^ 


X 


^T" 


167^ 


1_  __^_ 


,  etc. 


8. 


I  -^  I  (  I  +  *  —  i  +  tV»  etc.,  Ans. 


2  +  1)       I 

2+I-i)-i 


i-i  +  A 


2  +  1-I  + A) 


^  —  ^4?  ^^^c. 


10.     The  exponents  of  x  decrease  by  2.  (Art.  271.) 

The  coefficients  are  found  by  means  of  the  Binomial 
Formula.     (Art.  270.) 

ist  coef.  =  I 

2d      "     =n  =  \ 

n  ^^  __  i  X  —  I i_ 

2                       2  2«4 


3d 


<<; 


=  W  X 


.,        ,,  M—  IW—  2  I  % 

4th    "     =  7^  X X =  —  —  X  — 

■  2  3  2-4  3 


2'/\'(> 


IKI^I^ITE     SERIES.  189 


Sth  coei.  ^  71  X X  X 

234 

X  -^= ^5,  etc. 


•  2'4«6  4  2-4'6'8 

Hence  we  have, 

(:^  +  j,)i  =  r.  +  5:^  -  ?-'-i^'  +  3^""-!  -  3J^,  etc. 
^  ^'  2  2-4         2-4-6         2-4-6-8 

Or,  transferring  x  to  the  denominator  (Art.  279), 
^a;+2/j    -^+2^;       2.42:3+2.4.62:5       2.4.6.82:7'  ^^^• 

II.     Find  the  first  five  terms  as  in  the  preceding  sohition. 

,, ,    .                      n  —  I       71  —  2       n  —  X      71  —  4 
6th  term  =  ?j  x x  — —  x x 

2345 

_  ^'S      ^       i  _       3-5-7 


2.4'6.8  5        2.4.6.8.10 


rage  231. 

I.  The  ratio  may  be  found  by  dividing  any  term  by  the 
preceding  one. 

a  I  I 

s  = =  — y  =  „  =  li   Ans, 

__  I  _  ^  _  2     J 

I  I 


3.     s  =  — ^  =  Y  =  I,  Ans. 
1  ^^  -ij       -5^ 


2        2 

X  1 

_2 _    2    _     .1 


4.     5  =  ; 2  =  1=  ^h  ^^«» 


1  —  3         3 

_  _^_  _  *  _ 


5.      S  =  —^2  =  Y  =  2,  ^W5. 


6.  s  = ^  =  1=9,  Ans, 

4  4  ^ 

7.  S  = 3   =  2  =   10.  ^?i«. 


too 


LOG  ARITiiMg. 


8.      5  = 


9.     s  = 


3 

TO 


I  — 


TO 


3 
10 

9 
10 


=  -'^  =  h  Ans. 


6 
To 


6 
TO 


I  -TO 


I      —     9     —    3' 


9 
TT7 


2     ^^5. 


10.     S 


=  i^(,_i)^l^('i:=_L)^ 

a      \        a/       a      \    a    I 


T  flf 

-  X  — - — 

a      a  —  I 


II.     s  =. 


10 


4 


10 


— ,    ^;i5. 

a  —  I 

50  rods,  Ans, 


LOGARITHMS. 


rage  235. 

I. 

Log. 

7 

= 

0.84510, 

A71S, 

2. 

« 

9 

— 

0.95424, 

Ans, 

3. 

« 

108 

= 

2.03342, 

Ans. 

4. 

a 

176 

— 

2.24551, 

Ans. 

5- 

ce 

1990 

— 

3.29885 

Log. 

223   X  9 
1999 

: 

200 

3.30085, 

Ans. 

6. 

« 

0-95 

= 

1.97772, 

Ans, 

7. 

<( 

0.0125 

— 

2.09691, 

Ans. 

8. 

a 

0.0075 

— 

3.87506, 

Ans. 

9- 

a 

16.40 

"^^ 

1. 21484 

264  X   5 

— 

132 

Log. 

16.45 

= 

1.21616, 

Ans 

0. 

(( 

185.0 

— 

2.26717 

235   X   3 

— 

70 

Log. 

185.3 

— 

2.26787, 

Ans, 

LOGARITHMS.  191 

11.  Given  2.17231  z=z  log.  148.7,  Ans. 

17026 
301  )  20500  (  68 

12.  Given  1.2526 1  =  log.  17.89,  Ans. 

25042 
249  )  21900  (  87 

13.  Given  3.27715  =  log.  1893,  Ans, 

27646 
235  )  69000  (  293 

14.  Given  2.30963  =  log.  204,  Ans. 

15.  Given  4.29797  =  log.  19858.29,  Ans, 

29667 
223  )  1300000  (  5829 

16.  Given  1. 14488  =  log.  c.1396,  Ans, 

14302 
322  )  18600  (  57 

17.  Given  2.29136  =  log.  0.01956,  Ans. 

29003 

223  )   13300  (  59 

18.  Given  3.30928  =  log.  0.002038,  Ans, 

30750 
212  )   1780  (  8 

Page  237. 

2.     Log.  109.0        =     2.03743 

416  X  3  =  125 

'•     14.10        =     1. 14922 

301  X  7  =  211 

3. 1 900 1  =  loo.  1548.86,  Ans. 

18752 
281  )    249000  (  886 


193  LOGARITHMS. 


3- 

Log.  1.460 

.16435 

301  X5 

151 

"  1.340 

— 

.12711 

322  X7 

225 

.29522  —  log.  T.973,  Ans. 
447 
223  )  750  (  3 

4. 

Log.  .074 

^z 

2.86923 

"  1500 

= 

3.17609 

2.04532  =  log.  Ill,  Ans, 


6.  Log.  12.40   =   1.09342 
349  X  8  =     279 

1.09621 

*'     0.16    =   1. 20412 


1.89209  =  log.  78,  Ans, 


Log.  .041;    =   2.65321 
"  1.20    =   .07918 


2.57403  =  log.  .0375,  Ans. 


8.  Log.  1.380   =   .13988 
322  X  I  = 32 

. 14020 
"  .096    =   2.98227 


1.15793  =  log-  14-38,  Ans, 
534 
301  )  2590  (  8 


Log.  — 128  =   2.T0721 
"   —47  =   1. 67210 


.43511  =  log.  2.723,  Ans, 
457 

158  )  540  (3 


loCtAKithm^.  19^ 


lO. 


Log. 

186 

2.26951 

a 

—  0.064 

2.80618 

3-4^333   —  log-  2906.3,  Ans, 

240 

147  )  930  (  63 

Log. 

—  0.156 

zzz 

1-19313 

a 

—0.86 

1-93450 

1.25863   log.  .1814,  Ans, 
768 

235  )  950  (  4 

Log. 

—  0.194 

— 

1.28780 

a 

0.042 

— 

2.62325 

.66455  =  log.  —4.619,  Ans, 

370 
93  )  850  (  9 

rage  239, 

14.  Log.  .135  =      1. 13033 

4 

4.52132  =  log.  .0003321,  Ans, 

15.  Log.  1.42  X  10  =      1.5229    =  log.  33-335.  ^ns, 

.52244 


16.     Log.  1.230        = 
349  X  4    = 


130  )  4600 

(  35 

.08991 

140 

.09131 

25 

45655 

18262 

:  log. 

191 

-77. 

2.28275  = 

Ans, 

103 

223  )  17200  (  77 


104  LOGARITHMS. 

i8.     Log.  143.0        =      2.15534 

301  X  2      =  60 

3  )  2.15594 

.71865  =  log.  5.23,  Ans, 

19.  Log.  1.62         z=        .20952 

Dividing  by  6,  .03492  =  log.  1.0836,  J ?25. 

342 
416  )  15000  (  36 

20.  Log.  1540        =      3.18752 

281x9    =  253 

8  )  3-19005 

•39875  =  log-  2.504,  A^is. 
794 
171  )  810  (  4 

21.  Log.  1876 -T- 10   =   .32732  =  log.  2.124,  Ans. 

rage  240. 

23.     Log.  .001624    =      3.21058 
Div.  by  6,      6)64-3.21058 

1-53509  =  log.  .342  +  ,  Ans. 

24.      Log.  .01449        =r        2.16107 

Div.  by  7,      7  )  7+5.16107 

1.73729  =  log.  .546+,  ^^i5. 

25.     Log.  .0001236    =     4.09200 
Divid.  by  8,  8)8  +  4.09200 

1.5 1 150  =  log.  .324+,  Ans. 

27.     Log.  1.07  X  4   =      -11752 
«      1500         =     3-17^09 

Adding,  3.29361  =  log.  $1966.05,  Ans. 


feUSlKESS     FORMULAS.  19d 

28.     Log.  1.05  X33   =       .69927 

"'      370  =:       2.56820 

Adding,  ^       3.26747  =  log.  $1851.27  4-,^?25. 


BUSINESS    FORMULAS. 
Par/e  24^'>. 

2.  p  =  cr  =  $4370  X  .08  =  $349.60,  A71S. 

P  ^500  i.  A 

3.  r  =  -  = r=  .20 ;  20  per  cent,  ^;^s. 

^  c        2500  '        -^ 

$^00 
4-     ^'  =  ^^Yg^Q  —  -161;  16J  per  cent,  .4??5.    (Art.  238.) 

7;       I67.48        „    .  . 

5.  c  =  -  = =  $5269.92,  Arts, 

*»  r  .25  ^  ^ 

6.  c  =  '-  =  -^^  =  I12600,  Ans, 

r        .i2i 

7.  s  =  c{i  +  r)  =  $750  X  1. 15  —  1:862.50,  Ans. 

8.  s  =  c{i  —r)  =  ^960  X  .87I  =  $840,  ^7is. 

s  $S40         i^ 

10.     c  = =  -^^^^  =  I600,  A71S 

I  —  r        .90 

12.  ^  =  — -  =  16J;  16 1  years,  Ans, 

13.  ^  =  —  =  10;  10  years,  Ans. 

14-     **  =  7  =  o  =  'i^l;  12J  per  cent,  ^^25. 

15.     r  =  J  =  — ■  =  .02i;  2f  per  cent,  Ans. 

17.  a=z  2^{i  +  r)"  =  I1500  X  (1.05)6  =  -^2010.14,  ^;25. 

18.  a  =  $2000  X  (1.03)6  =  ^2388.05,  A71S. 

19.  a  =  ^5000  X  (i.oi)^  =  I5414.28,  A?is. 


196  BUSINESS     FORMULAS. 

„  s  I3600 

I  +  nr  1.30  /   V    v).  1  '  ;>  J;?s-. 

$3600  —  I2 769.23  =  I830.77,  discount, 

22.    P  = =  $6000,  present  worth  ;      /   , 

1.30  >Ans, 

cl  =  I7800  —  I56000  =  $1800,  discount,  / 

T1  S  I23OO        „       ^  . 

26.  tZ  =  I2500  X  .15  =  I375; 

I2500  —  $375  =  $2125,  cash  value,  Ans. 

c  (i  +  r)       $1.7';  X  1.20       . 
I  —  <:/  .90  ^^^ 


^j.qo  X  1. 21;        .„  ^ 

29.    m  =  — ^  =  $S.Ss  +  ,  Ans. 

.92 

^r         I6000  X  .06 

31.   M  =^—  =  ^2 T~ir^-  =  5MI  P^r  cent,  ^?Z5. 

*^  c        I6000  +  I180       *^^"^  -^ 


32.    E  —^=      l^^^  ^  '^^  =  i2i  per  cent,  A71S, 

C  ^1000  —  «|)200 


33-   -^  =  IsoooVs'/o"  ^  9A  per  cent,  ^«.. 
34.    R  =  — ^r '- —  =  6  per  cent,  Massachusetts; 

qpIOO 
„  $100    X    .08  .,  .     /%!  • 

Hence  the  Ohio  bonds  are  preferable,  Ans. 

s  $215000        .     ^ 

36.     a  = =  — =  $24630.54,  invest. ;    .    . 

^  14-r         1.015  t  o    j-tj  >Ans, 


$25000— $24630.54  =  $369.46,  commission 


(i+r)"— I  (1.07)*— I.  .3108  „ 

38.     a  =z  ^-I— ^ s  =  ^^ — -^ $300  =  -^ —  $300 

^^  r  .07         "^  .07 

=  $1332,  A71S, 


BUSINESS     FORMULAS.  197 

(l. 05)1*'— I   .  .629015,,  .  , 

39.  a  —  ^— ^^ $500= — ^ — ^•^500=16290.15,  Ans. 

(i  +r)«_  I  {1.05)5—  I 

40.  s  =  a-T-  - — -—^ =  $5000  -^  ^  ^^ 


*•       "        .05 

250  . 
=  —z  =  ^905.80,  Ans. 
.276    -'   ^ 

(i  +r)"—  I   .       (1.10)5  —  I 

41.  §  =  fl5  -i-  -i ^^ =  $20000  -r- 

r  .10 

=  I20000  -T-  .61  =  $3278.69,  Ans. 

(i.o6)i0— I   I1800   ,   00. 

42.  S   =  $30000-^-^^ — ■  = =  I2 278.48,  ^?2S. 

^  .06  .79 

(1.07)^ — I,,     ."iioS.,     „  ^„, 

43.  5  =  ^ — ^-^ '^650  r=  -^ $650  =  I2886,  Ans. 

.07  .07 

Note. — The  answers  will  vary  slightly  according  to  the  number  of 
decunals  used  in  the  solution. 

44.  ,  =  ('  +  '•)'  -  '  a  =  il£^'^  $880  =  :4i8  ^33^ 

r  .06  .06 

=  $6130.67,  Ans. 

(I.OO'^ I     ^  •407     >,  A.  ^  ^  A 

45.  ,*?  —  ^^ — ^ I340  =  -^-^$340  =  $2767.60,  Ans. 

.05  .05 

47.    ^   = ^ — ^^$525  =  --^%25  =  $2336.25,  Ans, 

.04  '04 

49.    P  =  -  =z  — ~-  =  S14166.67,  A71S, 
r        .06  ' 

51.    P  =  ^  [(i  +  r)-"  -  (i  +r)-'-^] 
P  =  ^-^[(i.o6)-e-(x.o6)-] 

.06  ^''^ 

_  Pr  _  $3840  X  .05 

53-     «^  -  I  _^+  r)--  ~  I  -(1.05P0 
$3840  X  .05 


.7687 


=  I249.77,  Ans, 


198  IMAGINARY     QUAIS^TITIES. 


IMAGINARY     QUANTITIES. 
rages  265,  206, 


2.  +  V—x  X  —  V—  y 

=  +  V^  X  V—  I  X  —  Vy  X  V—  I 
=  —  Vxy  X  —  I  =  +  V^y,  ^ns, 

3.  V—  9  X  a/—  4  r=  1/9  X  V—  I  X  A/4  X  V—  I 

=  6x  —  1  =  —  6,  ^>i5. 

4.  V—  2  X  a/iS  =  a/2  X  a/—  I  X  a/i8 


=  A/36  X  a/— ^i  =  6a/—  I,  Ans, 


5.     a/— a?  X  Vy    —  Vx  X  a/—  I  X  \^y 


=  a/^^?/  X  a/—  I  =  a/—  xy,  Ans, 


V —  X        \/x  X  a/ —  I  A 

7.     — ....^  =  — = j==^  =  I,  Ans. 

V  —  X      Vx  X  a/ —  I 


Vy  Vy  \     y 


a/cc               a/^  /  ^      4 

9.    — - —  =  — = —  =  \/  — '  ^^^^^* 

V—y      Vy  X  a/ —  I       V  ~  2^ 


ioa/—  14         io\^i4  X  a/—  I  /-      4 

10.    — ;=^  =  — 7^ 7=^=-  =  ^^2,  Ans. 

2a/—  7  2A/7  X  y  —  I 


cV  —  I       c      . 

II.       -;zr=   =   ^,    ^?Z6-. 


fh/-  T         ^ 


NEGATIVE     SOLUTIONS.  199 


IMPOSSIBLE     PROBLEMS. 
Page  20S, 

2,     Let  a;  =  the  number. 

Then  =  ik 

5       4 

Clearing  of  fractions,     4X  —  5:^  =  300. 
It  is  impossible  that  4X  should  be  greater  than  $x. 


3- 

Let 

X       one  part. 

And 

y       the  other. 

Then 

X  +  y  —  S 

And 

xy       18 

(0 

Now  the  product  will  be  greatest  when  the  parts  are  equal. 
Making  y  equal  to  x,  the  equations  become, 

2^=8  (3) 

:^^  —  18  (4) 

From  (3)*  X  =  4  (5) 

Squaring  (5),      a:^  =  16  (6) 

Equations    (4)   and   (6)   are   contradictory.     Hence,  the 
problem  is  impossible. 


NEGATIVE     SOLUTIONS. 

Page  209. 

Let  X  =  the  number  of  years. 

Then  36  +  :c  =  (20  +  x)  2 

Uniting  terms,  x  ^  —4,  Am, 


200  hor:n^er's    method. 


HORNER'S     METHOD. 


rffge  27 

s. 

3. 

^  +  3^^  +  5^  —  178. 

A 

B 

C 

D                  a    b  cde 

I 

+  3 

+  5 

= 

178   (  X  =  4.  5388,  ^yi». 

4 

28 

132 

7 

33 

46  =  !)'            B^G'=h 

4 

44 

42.375       46-^77  =.5 

II 

77=.C' 

3.625  =  D" 

4 

7.75 

2.797377 

15  =  5' 

84.75 

.827623  =  D" 

.5 

8 

.749942 

15-5 

92.75  =  C" 

.077681 

1 

•5 

•4959 

.074994 

16 

932459 

.5 

.4968 

16.5  =  £ 

V       93.74  27  = 

C" 

•03 

16.53 

•03 
16.56 

4. 

p;3   +  9.^-2  _ 

-  jx  -  2200. 

•. 

A 

B 

C 

D                abed 

5 

+  9 

-7 

— 

2200   (  a'=7. 1073536,  ^;is. 

35 

+  308 

2107 

44 

+  301 

93  =  1)' 

35 

553 

86.545 

79 

854  =  c 

6.455  -  B" 

35 

11-45 

6.144311215 

114  =  B' 

865.45 

.3106887S5  =  D" 

•5 

11.5 

.263570321 

114- 5 

876.95  =  C" 

.047118464 

.5 
115 

.808745 

.043928386 

877.758745 

.003190078 

•5 
115.5  =  B" 

.80899 

878.56717315= 

=(7 

.002635703 

'"  .000554375 

■035 

.C00527140 

115.535 

.000027235 

.035 

115-57 


TEST     EXAMPLES     FOR     IlEVIEW. 


201 


5.     2'3  +  x^  +  .r  =  100. 


A        B. 

C 

D           ahcde 

I    +1 

+  1 

=  100   (4.264429  + 

_4 

20 

84 

5 

21 

16  =  D' 

4 

36 

11.928 

9 

57  =  t7' 

4.072  =  D' 

_4 

2.64 

3.788376 

13  =  B> 

59-64 

.283624  =  U" 

13.2 

2.68 

.256071744 

134 

62.32  =  G" 

.027552256  =  i>»^ 

13.6  =  B" 

.8196 

.025631441984 

13.66 

63.1396 

.001920814016  =  iV" 

13.72 

8232 

.001281682442 

13.78  =  B" 

63  9628  =  C" 

.000639131574 

13.784 

-055136 

-000576757098 

13-788 

64.017936 

.000062374476 

13.792  =  fi«^ 

.055152 

137924 

64.073088  =  C'^" 

13-7928 

.00551696 

64.07860496 
.00551712 

64.084122  o|8  = 

iT  =  4.264429  +  ,  Ans. 

I. 

2. 


TEST     EXAMPLES     FOR    REVIEW. 

Prff/e  274:, 

6«  +  4«  X  5  +  8a-T-2  —  3«  +  i2«  X  4  =  75«,  Ans. 
(8.'C  +  3a:)5+4.^-f-7  — (5.^4-93;)  ^7  =  57a;+7,  ^/^s. 
(sax  —  «^  +  4c^)  —  {2ax  —  4^?^  +  2cd) 

=  3«'^'  +  3«^  +  2C(h  Ans. 
4bc  +  [T,cd  —  (2.?;?/  —  ??iM)  5  +  Tfic] 


202  TEST     EXAMPLES     FOE     REVIEW, 

5.  See  Book,  Article  75,  Prin.  4. 

6.  See  Book,  Article  91. 

7.  See  Book,  Article  91. 

8.  Given  £^  _  (a;  +  8)  =  ^  +  -  -  i7f 

3  9         7 

Uniting  terms,  —  —  x  =^  — ^  —  8 

3  3 

Multiplying  by  3,  etc.,  .t  =  8,  ^4;i6'. 

., ,  4^;^      X  -16       7.1 

9.  Given  ^ — -. \-  2x  —  ^-^  X  ^^- 

5532 

Reducinpr,  ()x  =  —  x  — 

3        2 

a*   =    31,    ^??,S'. 

10.  3^%  —  65V  —  c^Z  =:  c  (sb-  —  6I)'^c  —  cd),  Ans. 

11.  32%  —  gxh  —  iSx^yz  =  3^^(y  —  3^  —6yz),  Ans. 

12.  a2»  —  Z^2«  —  (^«  -I-  ^")  (<2'^  _  5"),  Ans. 

13.  8(7  —  4  =  2  X  2  (2<T  —  i),  Ans. 

14.  cf4  —  I  =  (^2  ^  i)  (f^  _|_  i)  (^  _  i)^  ji^is^ 

15.  Let  ic  =  one, 
Then                     31  —  x  =  the  other. 

And         5^  — 9(31— •'^)  =  I 
Eeducing,  142;  =  280 

X  =z  20  ;)    , 

}  Ans. 

31  —X  =   II,    ^ 

16.  See  Book,  Article  233. 

17.  Let  X  =  No.  shots  each  fired. 
Then                   ^x  +  ^x  =  34 

Eeducing,  Sx  +  gx  =34x12 

a:  =  24  shots,  Ans. 


TEST     EXAMPLES     LOK     KEVIEVV.  203 


rage  275, 

1 8.     Denote  the  quantities  by  x  and  y. 

Then  xy  —  a  (i) 

'-  =  ^  (2) 

From  (2),  X  ^  hy 

Substituting  in  (i),        hy'^  =  a 

Dividing  by  h,  y^  = 


20. 


Extracting  root,  ?/  =  ± 

From  (2),  y  =  I 


a 

h 


x^ 


Substituting  in  (i),  —  =  « 

Multiplying  by  h,  x^  =:  ab 

Extracting  root,  x  =  ±:  \/ab  ; 

And  3/  =  ±  Y  ^. 


A?is. 


«2  —  I        («  +  i)  (d!  —  i)        «  +  I       . 
19.     -^-T  =  -^^ — T-r— — ^ — ~  =  — r—  5  ^^i«- 


a  ■\-h 


b{a 

-I) 

I 

7,  > 

Ans, 

a^  —  6^       a  —  b 

21.  ()X^y^  +  i2a:?/2;  +  4^^  =  (32'^  +  2^)  {^xy  +  22),  J/^s. 

22.  952  —  6Jc  +  6*^  =  (3^  —  c)  (3^  —  c),  Ans. 

23.  See  Book,  Article  231. 

24.  Let  40;  =  length  of  fence. 
Tlien  X  =  Xo.  of  acres. 
Reducing  area  to  sq.  rods,       x^  =  iGox 

X  =1  160 
And  4x  =  640  rods,  A}is» 


204 


TEST     EXAMPLES     FOR     KEVIEW 


25.  Let 
Then 
And 

By  conditions, 

Multiplying  by  9, 

26.  Given 

Multiplying  by  i — x, 
Transposing, 


29 


30- 


2X  =  entire  distance. 
X  --■  I J  =  hours  of  ascent, 
a;  -i-  4I  =      "      "  descent. 

2X         2X 

3         9  ^ 

8ic  =  117 

2X  =  2gl  miles,  A71S, 
,       I  -\-x 

0 =z  O 

I  —  X 

h  —  hx  =z  I  -{-  X 
bx  -\-  X  =z  b  ^  I 


X  = 


27.  See  Book,  Article  103. 

28.  See  Book,  Article  104. 

(^2   _   ^2)    (^    ^    y) 


b+  I 


,  A?is. 


(x  +  y)  {x  -  y)  {x  +  y) 


(x^  +  2xy  4-  2/^)  {x  —  y)  {x  +  yf  {x  —  y) 

=  I,  ^4/^5. 

a'^  —  ¥  _  {a^  J^  h\{a^--W) 

(rt2  _  2ah  +  ^2)  («2  +  ^2)    -    («"_  If  (^2  ^  J2) 


a 


,  ^?i5. 


31.     Multiply  the  terms  of  the  second  fraction  by  i—c?. 


■a'' 


32.     Let 


i  +  rt^       J  —  2ct?-\-a^ 


$a^—a'^ 


I— a'' 


I— a' 


■«'' 


,  Aus, 


X  =  number. 


33' 


tjfl  ry*  ry* 

Then  x  -\ 1 -=  154 

Clearing  of  fractions, 

60a:  +  15-i'  +  i2:r  —  10^  =  154  X  60 

Uniting  terms,  77.^  =  154  x  60 

x  =z  120,  Ajis. 

Let  X  =:  amount  each  had. 

Then  a;  —  I30  =  2  (.r  —  1540) 

X  =  $50,  Ans. 


TEST     EXAMPLES     T  O  R     R  E  \'  1  E  W .  205 

Pafje  276. 

34.  Let  X  =  No.  of  hours. 
Then  24  +  Sx  =  distance  ship  sails, 
And                                     i2x  —        "       privateer. 
By  conditions,                     1 20;  =  24  +  Sx. 

X  =  6  hours,  Ans. 

^.  Sx  +  ru  .  . 

35.  Cliven  —J-      =  7  (U 

And  H^-^-y^o  (2) 

Clearing  of  fractions,   Sx-\-$y  =  4g  (3) 

And  —3'^' +  5//  =  o  (4) 

Mult.  (3)  by  5,  40^'+  151/  =  245 

"     (4)  by  3,         — 9^+15^  ^  Q 
Subtracting,  49a;  =245 

X  =  S>  I  ^^^^^ 
Substituting  5  for  x  in  (4),    ?/  =  3,   f 

^6.     Given  a;  =  ^^ f-  5  (i) 

And  4^ -—  =  3  (2) 

Clearing  of  fractions,    yx  —  i/  =:  ;^^  (3) 

And  —x-]-i2y=ig  (4) 

Mult.  (4)  by  7,      -  7^  +  84^/  =  133  (5) 

Adding  (3)  and  (5),  83?/  =  166 

Substituting  2  for  y  in  (i),  a;  =  5,  (  " 

37.     Let       a  =  the  distance  ;   m  =  rate  of  one; 

n  =  rate  of  the  other  ; 
And  X  =  the  required  time. 

Then         inx  •\-  nx  ^  a 

.     .*.        X  =  — — — .     Hence,  the 
m  +  11 

Rule. — Divide  the  given  distance  hy  the  sum  of  the  rates  ; 
the  quotient  will  le  tke  time  of  meeting. 


206 


TEST     EXAMPLES     FOE     REVIEW. 


38.     Let 
Then 

And 


39- 


40. 


X 


By  conditions, 
Multiplying  by  16, 


Let 

Then 

And 


12  +  4 

X 

12—4 
16  "^8 

X  +  2X 

SX 

',  2X 

X 

y 

i»  —  6 

X  -\-  6 


2X  =  entire  distance, 

time  down  stream, 


a 


up 


(( 


=  8 


2/  +  6 
Clearing  of  frac,  etc.,  2X  —  1/  = 
And  4X  —  sy  = 

Mult.  (3)  by  2,  4X  —  2ij  z= 

Subtracting  (4)  from  (5), 
Substitut.  18  for  ^  in  (3), 


Hence, 
Let 

Then 
And 

From  (i), 


y  = 

X   = 
X  

y~~ 

X  = 

y  = 

x-\-  y  \  ij    : : 
^2  _  ^2  — 


=  T28 
=  128 
=  85!  miles,  Ans. 

=  the  fraction, 

I 

2 

^  3 

4 
6 

—  6 

12 

18 
12 


a; 


2Sr 


12       . 


the  greater, 

"    less. 

8  :  3 
49 
5|/ 

3 

49 


Sub.  value  of  x  in  (2),  — - — /  = 

i6?/2  =  441 
4//  =  21 
//  =  -V-=5iless;  ) 
Sub.  -y  for  2/  in  (3),  2:  =  ^r^SJ  gr.,    \ 


Multiplying  by  9, 
Extracting  root, 


0) 

(3) 
(4) 
(5) 


(0 

(3) 
(4) 


Ans. 


TEST     EXAMPLES     FOR     REVIEW.  207 


41, 


42. 


43- 


Given 

I02'  +  6y        76 

(I) 

4?/  —  2Z               8 

(^) 

And 

6x  +  8;2  -  88 

(3) 

Multiplying  (2,  by  (4),         i6y  —  8^—32 

(4) 

Adding  (3)  and  (4), 

6x  +  161J        120 

(5) 

Multiplying  (5)  by  5, 

302'  +  80^        600 

((') 

(i)by3, 

2,ox  +  i8_^        228 

Subtracting, 

62y  —  372 
y  -(>]) 

Substituting  6  for  y  in  (i),               x       A-'->\  A\ 

US, 

«           6   "  ^  ' 

'  (2),                ;2       8,  ) 

Given 

2:t'  -{-  sy  +    z        24 

(I) 

30;  +    ?/  +  22        26 

(2) 

And 

.T  H-  2^/   +  32   —   34 

(3) 

Multiplying  (i)  by  3, 

62:  +  9//  +  32   _    72 

(4) 

(2)  by  2, 

6x  +  27/4-42  —  52 

(5) 

Subtracting  (5)  from 

(4),       72/  —    2  _  20 

(6) 

Multiplying  (3)  by  3, 

3.!'  +  6^  +  92        102 

(7) 

Subtracting  (2)  from 

{7),     sy  4-  7^  =   76 

(8) 

Multiplying  (6)  by  7, 

492/  —  72;  =  140 

Adding, 

541/            _  216 

^  —  4;) 
:n(6),                0-8;U; 

Substituting  4  for  y  \ 

ns. 

"          these  values  in  (i),      x       2,  ) 

Let 

ic  —  A's  money, 
y  —  B's      " 

And 

z  -  C's      " 

Then 

x-\-  y  +  z  —  $180 

(0 

a;  —  ^  +  2;       I60 

(2) 

And 

.-c  +  ^  -  2  _  - 

(3) 

Subtracting  (2)  from 

(l),           2y   —   't^I20 

?/       $60 

208  TEST     EXAMPLES     FOR     KEVIEW. 

Adding  (i)  aud  (2),       2X  -^  2Z  =  I240  (4) 

"        (2)     "    (3),         2X-^=     $60 

4 

Subtracting,  —  =  $180 

4 

z  =  I80,  C's;  ) 

Subst.  I80  for  0  in  (4),  x  =  I40,  A*s ;  >■  Ans, 

y  =  %o,  B\  ) 

44*     Let  X  =■  circumference  of  fore-wlieel, 

Then  x -^  s  =  "  "  hind    " 

A        J  240  240 

And  -—  z=z  — \-  40 

X  X  -\-  $ 

Clearing  of  fractions,  etc., 

x^  -\-  ^x  =  iS  * 

Comp.  sq.,  etc ,    a;  =  3  meters,  circ.  f.  wheel;  |    , 
And  x-j-  s  =  6      ''         "    h.    "        f 

45.     Let  X  =  side  of  one. 

Then  x  -\-  2  =    "     "the  other. 

Difference  of  cubes, 

6x^  4-  1 2:^'  +  8  =  488 
Eeducing,        x^  +  22^  =  80 
Comp.  sq.,  etc.,  x  =    8  ft.,  side  of  one ;    )    . 

And  a:  +  2  ==  10  "        "      other,  j 


Page  277. 

46.  See  Book,  Article  232. 

47.  Let  jx  =  one  part," 

And  iicc  =  the  other. 

Then  I8.^'  =  126 

.'.         x  =  J 

}  A71S. 
no:  =  77, 


TEST      EXAMPLES     FOK      REVIEW.  209 

48.  Let  X  =  number  of  meters. 

rm  ^120  $120 

Then  h  -So  = 

X  ^  X  —  ^ 

Dividing  by  .50,  clearing  of  fractions,  and  reducing, 

0?  —  S.r  ==1920 

Comp.  square,  etc.,  a;  =  48  meters,  Ans. 

49.  See  Book,  Article  472. 

s  =  c(i  +  v)  =  $175  X  1.25  =  $218.75; 

$218.75 -^  89  —  $2,457,  A)is. 

50.  See  Book,  Article  490. 

51.  Let  a;  =  price  of  horse, 
Then                    a;  +  ^100  =        "       carriage. 
And                  a:+ioo  :  x    ::    x  :  $0 
Changing  to  an  equation, 

X^  —  ^OX  r=  5000 

Comp.  square,  etc.,  x  =  $100,  horse; 

And  X  4-  $100  =  $200,  carriage. 


Ans. 


52.  See  Book,  Article  247. 

53.  Let  _x  —  share  of  younger, 
Then          a;  +  35  =        "       elder. 
Adding,     2X  +  35  =  165 

Transp.,  etc.,        x  =    65  hectares,  younger;  ) 

a;  +  35  =  100       "         elder,'        \  ^^^^' 


4- 

Let 

X       the  number 

Then                   3a; 

X 

—  40  _  5 1 

2 

Transposing,  etc., 

7.r        182 

• 
•  9 

X  _  26,  Ans. 

•210  TEST     EXAMPLES     FOR     REVIEW. 

55.  Let  X  =z  price  of  a  sheep. 

?/  =     "        "    lamb. 
Then  6x -{-    ry  =  $ji  (i) 

4X  -^    8i/  =  1^64  (2) 

Mult.  (2)  by  6,  24^-f  "4%  =  ^384 

"      (i)  by  4,  24ic  +  281/  t=  ^^284 

Subtracting,  2oy  =  lioo 

Substituting  5  for  ^  in  (2),      x  =  $6,  \ 

56.  Ijet  X  =  No.  who  voted  for  one, 
Then         0^4-271=    "  "         "      the  other. 

And        2X  -h  271  =  1425 
Transposing,      2X  =  1154 

:c  =  5  7  7  for  one ; 
And  a;  4-  271  =  848    "  the  other, 

57.  Let  X  =  C's  age, 
Then                 3a;  =  B's   " 
And                   6x  =  A's  " 

Adding,  102;  =  150 

X  =  1$  years,  C's  age; 
^x  =  45      "      B's    "     I  Ans. 

6x  =  90     "     A's    " 

58.  \/243  =  V81  X  3  =  gVs,  ^^is. 


59- 


Ans. 


^Jif  +  af-  =  Vy'(i  +  a)  =  yVi+a,  ^ns. 


Page  278, 
60.    x^  =  (x^)^;    y^  =  {y^Y,  Ans, 


3/- 


61.  3  (f/  —  &)  =  V  27  {a  —  ^)^  ^w."?. 

62.  Let  X  =:  price  per  bushel. 
Then     I'jx  —  it^x  =z  $3.60 

Or,  4X  =  I3.60 

iv  =  $0.90,  ^4?^6'. 


» ♦ 


TEST     EXAMPLES     FOR     REVIEW. 


211 


^3' 


64. 


Let  X 

Then  x  -\-  S 

And  2x  4-  8 

Transposing,       x 

X  +  8 


amount  lost  on  second, 

first. 


i( 


(S 


2r<- 


$  I  o  on  the  second ;  \     . 
$18      "       first,        f 


71S. 


I 


3 

4 


Let  X  =  men  first  employed. 

Then,  by  conditions,     6x  :  io(.?;-|-i2)    ; 
Changing  to  an  equation,  etc.,     8;r  =120 

a*  =  15  men,  Ans. 


65. 

Let 

X       number  in  the  party. 

Then 

Sx       amount  they  paid. 

And 

Sx  —  7  (x  4-  4) 

Eeducing, 

a;  =  28,  Ans. 

66. 

Let 

X       wages  of  a  woman. 

And 

«/  —      "        "    boy. 

Then 

Bx  +    6y  —  $72                      (i) 
6x  -\-  iiy       »^8o                      (2) 

Mult.  (2)  by  4, 

24a;  +  44_^        $320 

"      (Obys, 

24.T  4-  18^  —  $216 

Subtracting, 

26^        I104 

•••        y  -  ^4,  boy;       \  ^^^^ 
fovym{i),  x       $6,  woman,  f 

Substituting  4 

• 

Given 

67. 

X  i7x       3^/17^'.    Hence,  9,  ^ws. 

68. 

V'-C  +12           V«  +   12 

Squaring,  etc., 

;r       a.  Ans. 

69.     Given 

Multiplying  by  Vy,  etc.. 
Factoring  and  dividing,       y 


a/// 

y    ((y 

«/ 

Vy 

I 

y    «i/ 

y 

I 
I  —  a' 

^  ;?<s'. 

212 


TEST     EXAMPLES     FOR     REVIEW 


70. 

Given 

^/x'- 

-  ^al)       a  —  h 

Squaring, 

x^  - 

-  4ab       a^  —  znh  +  h^ 

Transposing, 

x^       a^  +  2ah  +  IP" 

Extracting  root, 

X  —  a  -\-  0,  Ans. 

71- 

Let 

X       capacity. 

Then 

X 

3 

X 

+  40    - 

• 
•-  • 

X       240  liters,  Ans, 

72.  See  Book,  Article  238. 

73.  Denote  the  parts  by  x  and  i/. 
Then  x  -{-  y  =^  20 

And  x^  :  'ip    ::    4:9 

Changing  to  an  equation,  gx^  =  4^^ 


Extracting  root, 


Subst.  in  (i), 

Multiplying  by  3, 
Or, 

From  (3), 


21/ 


SX  =  21J 


+  ij  —  20 


2?/  +  3^  =  60 

sy  =  60 

y  —  12;)     . 
X  =z    8,  ) 


(3) 


74.     Let 
Then 

Subtracting, 
And 

Subtracting, 
Dividing  by  7, 
Multiplying  by  9, 


Page  279. 

'J2X 

Sx 
64a; 
_8x 
S6x 


Sx  = 

72^  = 


amount  at  first, 
tirst  payment, 
remainder, 
second  payment. 

'^140 

1 1 80,  Ans, 


TEST     EXAMPLES     FOR     KEVIEW.  213 

X 

7c.     Let  -  =  tlic  fraction. 

^  y      • 

The  -^   =  -  (i) 

And  — ^—  =  -  (2) 

2^  +  2        3 

Clearing  of  fractions,  4^  +  4  =  3//  +  3  (3) 

And                                       ^x  —  2y  -\-  2  (4) 

Transposing,              4^  —  3?/  =  —  i  (5) 

And                             3.-^  —  2y  —  2  (6) 

Mult.  (5)  by  2,           8a;  -  67/  =  -  2  (7) 

''      (6)  by  3,           9^  -  6?/  =       6  (8) 
Subtracting,                 x            =8 
Subst.  8  for  a;  in  (4),             y  =     11 

Hence,  -  =  — ,  Ans, 

y       II 

76.  Let  X  =  No.  of  dollars  watch  cost. 

X 

Then  -^  =  per  cent  rained. 

100       ^  ^ 

And  =  percentage  gained. 

By  the  problem,        ~ \-  x  :=  %\'ji 

''  100 

Mult,  by  100,  x^  +  loox  =:  17 100 

Comp.  sq.,     .^'■^-|- iooa;H-25oo  :=  17100  +  2500 

Extracting  root,  ^  +  50  =  ±  V17100  +  2500 

Transposing,  x  =  —  50  ±  a/ 19600 

Reducing,  x  =:  — 50^140 

X  =  $90,    A71S. 

77.  Let  X  z=  sum  each  receiyed. 

Then  x  +  15  =  i^^  —  9)  3 

Or,  X+  IS  z=z  ^x—  27 

Transposing,  2.'c  =  42 

X  =  $21,  Ans, 


214 


TEST     EXAMPLES     roll     REVIEW. 


78. 

Let 
Then 

And 

X       number  of  sheep. 

I120            .            .      ^ 
pnce  per  head. 

X 

I120         I120    '    ^ 

—       .    ^  +  f  I 
X          X  -\-  6 

Clearing  of  fractions, 

i2o:z:  - 

4-720        120.^::  4-  a;2  -f  6x 

Transposing, 

x^ 

+  6x        720 

Comp.  sq.,    x^ 

-\-  6x  -\-  g        720  +  9 

Ext.  root,  etc.. 

• 
•  • 

X  —  —3-27 
X       24  sheep  ;  ) 

120            .               ,         >  A71S. 

—       I5,  each, 
24                         ; 

And 

79- 

Let 

lox  +  y       number. 

Then 

iox-{-y  -  9i^'  +  U) 

(I) 

And 

iox-\-  y  —  6:^  —  loy  +  x 

(2) 

From  (i), 

X  -  8y 

(3) 

"      (2), 

gx  —  gy  =z  63 

(4) 

Dividing  (4)  by  9, 

x  —  y  -_  7 

Substituting, 

y  -  1 

From  (3), 

X  —  d> 

Hence, 

loa;  -{-  y  —  81,  Ans» 

80. 

Let 

3^ 

—  distance  one  goes. 

Then 

'JX 

"        the  other  goes. 

« 

And 

102; 

—  150  miles. 

• 
•  • 

X 

3^ 

^x 

-  15     " 

-  45  miles,  one;    )  ^^^^ 
.  -  105      "      other,  j 

TEST     EXAMPLES     FOTl     liEVTEW.  2l5 


8r, 

Let 

X  —  Xo.  of  B's  acres, 

Thou 

X  ^  lo  —      "        A's     " 

And 

$2800                     ,                                         „  -r,, 

— — -  _  price  per  acre  of  B  s, 

X 

a 

^2800           ^,       ,,      ,,         ^,^ 
a;  +  10 

tt 

2800         2800 

X            X  4-  ^o 

Clear,  of  frac,     28002;  +  28000  =  2  8oo.T-f-5:?:2-}-5oa: 
Transposing,  etc.,  x^  -\-  lox  =:  5600 

Completing  square,  etc.,  x^  —5  +  ^/5600  +  25 

Reducing,  x  =  —  5  ±  7  5 

x  =  'JO  acr.  B; 
And  a;  +  10  =  80    "  A, 


Ans. 


82.     {Vx  +  V7)  ( V'^  —  V7)  =  ir  —  7. 
Hence,  a/^'— V7?^'^^' 

^Z'    { Vs^  -  V3^)  ( Vs^  +  Vs^)  =  3-^'  -  zy- 

Hence,  V^  +  VzU'  -^  ^^■^' 

84.  Given  V^V^  =  _-£±.A^ 

Multiply,  by  denom.,    Z^  +  v^  =  r?  +  3 
Transposing,  ^/^^'  =  rZ  +  3  —  //^ 

Squaring,         x  =  d^-i-6d— -21^(1 -\-g  —  6b^-\-¥,  Ans. 

85.  Let  iP  =  clerk's  salary. 

Then  loa;  =  mayor's  ^' 

And  no;  r=  1^13200 

.r  =  I J  200,  clerk;    )    . 
And  10.T  =  I12000,  mayor,  ) 


216 


TEST     EXAMPLES     FOR     REVIEW. 


S6.     Denote  the  numbers  by  x  and  y. 

Then  x  -{-  ij  \  x  —  y    ::    8:6 

And  X  —  //  :  xy    : :    i  :  36 

By  Theorem  i. 

And 

Sub.  7?/  for  ^  in  (2), 

Dividing  by  y. 


Equating  (i)  and  (3),      x 


X  =z  ^y 
xy  =  ^6x  —  s^y 
iy^  =  252?/  — 36?/ 
jy  =  216 

=   216,    ) 


(0 

(3) 


87. 


88. 


rage  280, 

Denote  the  numbers  by  x  and  y. 
Then  xy 

And  x^  —  y^  :  {x  —  yY 

By  Th.  6,   x^-\-xy-\-y'^  :  x^—2xy  +  y^ 
By  Theorem  8,  3:2:?/  :  (x  —  yY 

"        "  6,  iT?/  :  (:r  —  ?/)^ 

Vahie  of  xy  from  (i),     4S  :  {x  —  yY 
By  Theorem  6,  4  :  {x  —  yY 


a 


a 


(I 


a 


12, 


Combining  (i)  and  (3), 
"        (3)    "     (4), 


x  —  y 
x-y 

x  +  y 


48  (i) 

37  :  I  (2) 

37  :  I 

36  :  I 


12  :  I 
12  :  I 
I  :  I 


I  :  I 

(3) 
14  (4) 


iK  =  8;    «/  =  6,  .4^Z5'. 


ic 


Let  a;  =  price  per  dozen ;        —  =  price  of  one, 


And 

By  conditions.. 


X        144 
12-2 =  -— 

12  X 

144 


:=  No.  for  1 2  cents. 


Dividing  (i)  by  2, 


J  44 

X  -{-  I  . 

72 


+  2      (I) 
+  1      (2) 


Clearing  of  fractions,     72:^  +  72 
Transposing,  x^  -\-  x 

Completing  square,  etc.,  x 

r  —  —1-1-  U. 
»t-  —        2  ni   2 


Reducing, 


x  +  I 

72a::  +  .^2  +  a: 

72  

-  i  ±  V^!^ 
8  cents,  Ans. 


TEST     EXAMPLES     POR     REVIEW.  5i]'- 

89.  Let  X  =  Xo.  of  clavs  tliev  travel. 
Then                              8^  =  distance  one  goes, 
And                               jx  =         "       other  ^' 

By  the  problem,  15a;  =  150  miles, 

X  =z  10  davs,  Ans. 

90.  Let  X  =z  A's  income. 
Then                            3:^  =  B's      " 
And                              4x  =  I1876 

.-.        Of  =  $469,  A;  I  ^^^^ 
And  $x  =  $1407,  B,  j 

91.  Let  X  =  cost  of  cow. 
Then                            4X  =       "      horse, 
And                             ^x  =  I250 

X  =  $50,  cow;      I 
And  4:?;  =  I200,  horse,  j 

92.  Let  X  =  rate  per  hour  he  rode, 

24 
Then  -^  =r  hours  spent  in  riding, 

X 

And         -2^  :r=  8  =      «  «        walking. 

By  the  problem, 

24 

—  +  8  =  12 

a; 

ic  =  6  miles  per  hour,  Ans, 

93.  Let  X  z=z  the  length, 
And  ^  =    "    width. 

Then  2a;  +  2?/  =:  320  (i) 

And  ./•//  =  6000  (2) 

Dividing  (i)  by  2,      a;  +  ^  =  160  (3) 

Squaring  (3),    x^-\-2x?j-]-y^  =  2^600  (4) 

Mult.  (2)  by  4,         4:r7/  =  24000  (5) 

Subtracting,      x^—2xy-\-y'  =  1600  (6) 

Extracting  root,  x  —  //  =  40  (7) 

Combining  (3)  and  (7),     .r  =  100 ft. length;  )  ^ 
And  ^  =  60  ft.  width,    j 


318  TEST     EXAMPLES     FOR     REVIEW. 


04.     d  = =  — ^-^  =  44-     Hence  tlie  series 

^^  m  +1  6  ^ 

3^  lh  1 2  J,  17,  2 1  J,  26i   31,  ^?i5. 
95.      ?  =  «^  +  (^^  —  i)  fZ  =  1  +  49  X  i  =  25  ; 

«   +    ^  4+25  ^  T  ^ 

S  = X  ?i  =: X   50   =   637I,    ^^25. 


96.     Let  2:  =  No.  pair  bought;    r<?  —  5  =  No.  pair  sold; 

A  1  $100  .  .    - 

And  ■ =  price  paid  a  pair, 

~  =      "     received  a  pair. 

X  —  $ 

100         135 


By  the  problem, 

Reducing,  etc.,  ic  =  50  pair,  Ans, 

97.     Let    X  =z  one;    y  =  other. 

Then  x  :  y    ::    y  :  g         (i) 

And  y^  —  x^  —  128  (2) 

Changing  (i)  to  an  equation,      gx  =  jy 

■•■    ^  =  'f        (3) 

Substituting  in  (2),  y'^  —  -~-  =  128 

ol 

Multiplying  by  Si,        Siy^  —  49^/2  =  128  x  81 

?/2  z=  4  X  81 

Extracting  root,  y  =  18;)    . 

Substituting  18  for  «/ in  (3),         r?;  =  14,  f 

98      Let  X  =  No.  in  the  height. 

Then  x  +  43  =    "    "    "    length. 

And  x^  +  43a;  =  2400 


Comp.  sq.,  etc.,        x  =  —  ^  ±^2400  -f  u^^a 

Reducing,  x  ^=  —  ^-  ±  -^- 

a;  z=  32,  No.  in  heioht:  )     . 

r  A.7 
And  a;  +  43  =  75,    "    "  length,  ) 


*Tt:sT   i)5AMpLEs   fou   review.         210 

99.     Let  X  =  Bertha's  age, 

Then  ^x  =  Mothers  age. 

And  a;  +  20  =  (3^-  +  20)  | 

Reducing,    52;  -j-  100  =  9a;  +  60 

a;  =  10  years,  B.  ;  )     . 
2,x  =  30     '^      M.,  S 

(00.     Let  a;  =  price  i)aid  for  car. 

Then  —  =  what  each  would  have  paid, 

X 

And  —  =      *•        "   did  pay. 

X  X 

By  the  problem,     — •  = h  $i-75 

X  =  $105,  Ans, 

Page  281. 

loi.     See  Book,  Article  247, 

t=ah^=Wx  8=ff  =18^  hours =8:43-j^ o'clock,  ^1?Z5. 

102.  Let    X  =z  A's  pages;    y  =  B's  pages. 

Then  x  -\- tj  =  s7o  (i) 

And  3^  +  5^  =  2350  (2) 

Mult. (i) by 3,  s^-^sy  =  171Q  (3) 

Subt.  (3)  from  (2),  2y  =  640 

2j  =  320,  B's  pages;  I   ^^^^ 
From  (i),  X  =  250,  A's      '•       )  ^      ' 

103.  Let  X  =L  No.  of  days  it  will  last  the  man, 

Then  -  =  what  the  man  drinks  in  i  dav, 

X  "' 

^Q  z=         *'       wife         "         I  day. 

By  the  problem,        -^4 =  i 

*^       /  ^30 

Reducing,  etc.,  a;  =  20  days,  Ans. 

104.  See  book.  Article  476. 

i  —  ~  =2  —  =1  142  years,  Ans. 
r      .07         ^  " 


220  TEST     EXAMPLES     FOR     REVIEW." 

105.  Let  X  =  sum  in  the  purse. 
Then      3  +  2:7  +  2    : :    x  :  x  ■\-  2^ 

Or  5  :  9    : :    .^•  :  cc  +  24 

By  Theorem  i,        $x  -\-  120  =z  gx 

X  =  I30,  Ans, 

106.  Let  X  =z  whole  stock, 
And              f:r  —  200  =  A's  stock. 

Then  x  :  ^x  —  200    : :    3000  :  1600 

-n    mi              ^               5^  —  1800  . 

By  Theorem  6,       x  : ::    30  :  10 


9 

—  i8oo\ 

10 


By  Theorem  i,      i6x  =  ( J 

Multiplymgby  3,  482;  =  50a;  —  18000 

X  =  I9000,  whole  stock  ;  ^ 
And  fx  —  200  =  I4800,  A's         "       y  Ans, 

9000  —  4800  =  $4200,  B's         "       ) 

107.  Let  X  =  No.  votes  for  one, 
Then          a;  +  i  =        "        "       the  other. 

And  2:?;  +  I   zrr  369 

a;  =  184  votes  for  one  ;  )    . 

>  A71S 
And  3^  +  I  zi:  185      "      "    the  other,    j 

108.  Let    X  =  time  by  one ;    y  =  time  by  the  other. 
Then  -  =  part  one  does  in  i  day, 

X 

-  =     "    the  other  does  in  i  day. 

Since  both  together  can  do  the  whole  work  in  1 6  days, 
they  do  J  of  it  in  4  days  and  -^  of  it  in  i  day.     Hence, 

By  the  conditions,  -  +  -  =  —  (i) 

•^  X      y       16  V  / 

And  36  ^  3  /^x 

y      A 

Dividing  by  3,  etc.,  ?/  =  48  days. 

Subst.  48  for  y  in  (i),     ^  +  ^  =  ~ 
/.    X  =  24  days,  one ;    y  =:  4S  days,  other,  AnSo 


'      TEST     EXAMPLES     FOR     REVIEW.  221 

109.     Let  X  =  digit  in  tens'  place, 

And  y  =     *'      "  units'    *' 

Then  loa:  ^-  y  =  number. 

By  the  conditions,     — '       ^  =  ^i  (0 

And  lo.T  4-  y  +  9  =  lo?/  +  a;  (2) 

From  (i),  40a;  +  4/y  =  9.^7/  (3) 

Uniting  and  dividing  (2)  by  9, 

xi-  I  =  7J  (4) 

Substituting  value  of  ^  in  (3), 

40a;  +  4a;  +  4  =  gx^  +  9a; 
Transposing, 
Dividing  by  9, 

Completing  square,  etc.. 

Reducing, 
And 

From  (4), 
Hence, 

....  .  =  0=  =  iff  =  r,rs  =  s. 

Hence  the  series,  2,  6,  18,  54,  162,  486,  Ans. 
III.     Let  ■  a;  =  No.  of  hats. 


x^  — 

•35^'^'  —  4. 

35  y.                 4 

^-«±^ 

-r  —   3  s    .1 .  37 
•*^           18   i   18 

a;       4 

y  =  5 
+  ?/  _  45,  ^W5. 

1  etc. 

'4       1225 
9  "^    i82 

• 
•  • 

loa; 

'1369 
i82 

Then 

80 

X 

price  paid  apiece. 

And 

80 

X 

80 

h  I 

.r  +  4 

Clearing  of  fraction 

s,  80a' +  320 

So.r  +  a'2  f  4a: 

Transposing, 

etc.,           X 

320 

Completing  square, 

2  +  A/320--4 

Reducing, 

X 

?  +  18 

f  • 


a;  :^  16  hats,  .i?k^. 


222  TEST     EXAMPLES     FOR     REV^IEW. 

112.  ?  =  «r"~^  =  2x5^^  =  97656250,  Ans. 

113.  Let  X  =  length  of  shortest. 
Then            5:?;  +  3^  +  ic  =  90  feet. 
Uniting  terms,             gx  =  90    " 

r?;  =  10  feet; 

A71S. 


114. 


3^  —  30    " 

And 

Sx  —  50    " 

Let 

3^  +  19  —  3d, 

Then 

2X           2d, 

4a;  +  II          ist 

And 

9a;  +  30   _    219 

Transposing, 

gx        189 

X           2J 

4:?;+  II        95,   ist; 

2X  —  42,    2d; 

And 

3X  +  19  _  82,  3d, 

Ans, 


115.     Denote  the  numbers  by  x,  \/xy,  and  y. 

Then         x  +  \/xy  +  ^  =  14  (i) 

And  a^  +  a;«/  +  ^^  =  84  (2) 

Transposing  and  squaring  (i), 

x^  +  2xy  +  y^  =  196  —  28Vxy  +  a;^  (3) 
Subt.  (2)  from  (3),      rr^  ==  112  —  zSVxy  +  .-c^ 

Transposing,  etc.,    V^y  =  4  (4) 

Involving,  xy  =:  16 

And  3x1/  =  48  (5) 

Subtracting  (5)  from  (2), 

x^  —  2xy  -1-  ?/2  =  36 
Extracting  root,     x  —  y  =  ^  (^) 

Subst.  (4)  in  (i),    X  +  y  —  10  (7) 

Comb.  (6)  and  (7),        x  ^z  S 

y  =  2 


Vxy  =  4 
Hence  the  numbers,  8,  4,  and  2,  Ans. 


TEST     EXAMPLES     FOE     REVIEW.  223 

ii6.     Let  X  =  time  the  coffee  would  last  the  wife. 
The  man  drinks  i  lb.  in  4  weeks. 
His  wife        "       1   "     "  x     " 

Hence  he       "       -  "     "   i  week. 

4 

And  she         "      -  "     "   i      " 

X 

He  "      -  "     "3  weeks. 

4 

She  «      -  "     "  3      " 

X 

And  by  the  problem,        ^  +  -  =  i 

Clearing  of  fractions,    3:^  +  1 2  =43: 
Transposing,  a;  =  1 2  weeks,  Ans. 

117.  Let  X  =  sum  in  ist  purse, 

y  =z    "     «  2d      " 
Then  x  +  y  =^  ^300  (i) 

And  ic  —  30  =  ^  +  30  (2) 

From  (2),  X  —  y  -^  60  ^         (3) 

Subst.in(i),    y  +  6o-\-y  =  soo 
Transposing,  2y  =  240 

y  =  $i2o,  2d;)^^^^^ 
From  (3),  X  =  !i5i8o,  ist,  ) 

118.  Let  X  =  cost  of  the  cloth. 

Then  =  rate  per  cent. 

100 

X 

X  X  —  =  percentage  gained. 
100 

And  X  -{ =  39 

100 

Clearing  of  fractions, 

x^  +  IOO.Z;  =  3900 


Comp.  sq.,  etc.,  x  =  —  50  ±  ^3900  -t-  2500 

Reducing,  x  =  —  50  ±  80 

X  =  830,  Ans* 


224 


TEST     EXAMPLES     FOR     REVIEW. 


119.     Let 

And 
Then 

And 


122. 


X  =  No.  of  acres  in  one  part, 

2/  =       '^  *'      the  other  part 

X  -\-  y  =z  100  (i) 

202;  +  30^  =  2450  (2) 


Mult,  (i)  by  20,  20.T  +  20?/  =  2000 
Subtracting,  etc., 


From  (i), 


?/  =  45  acres;       , 


ic  =  55 


120. 

/  =  ar"~^  = 

:    2X3^^; 

Jr  —  a 

s        — — 
r  —  I 

2X3^^- 
~               2 

14348906,  Ans, 

121. 

Let 

X       No.  of  pine, 

And 

y  —    ''      "  hemlock. 

Then 

X 

+  ?/  -  300 

a;2  :  ?/- 


25  :  49 


Product  means  =  product  extremes, 
Extracting  sq.  root,         7:?;  =  SV 


7 

Subst.  in  (i), 

5?/ 

;VIult.  by  7,  etc., 

12^        2100 

From  (i), 

Let 

By  conditions, 

Reducing, 

Transposing, 


y  =  175.  ^'^"^^^^^^l^w.. 
X  =i  125,  pine,  f 

X  =  No.  of  men. 

2X  =  3i(-*  —  150) 
3^  =  5'^—  750 

2X  =   750 

o;  =  375  n^en,  ^?i5. 


APPENDIX. 


I.  Given. 

2. 


CUBE     ROOT. 
Page  284:, 


fl3 


3«2  )  3^2^-1-3^5^24-^3 


x^-\-6x^-\-i2x-\-2>  (  x-\-2,  Ans. 


x^ 


3^'  )  6.^•2-|-I2:^:-f-8 
3.^'2  -f-  62;  -f-  4  )  6.T~  4-  T  20;  4-  8 


3^;^  )  —  6:?;2^  4-  i  zxy"^ — Zif 


5' 


i2a^ — 2 


6. 


8ft3_48flr2_|_^6^_54  (  2rt— 4,  Ans, 
W 

i2«2  )  —  48rt2  4-96r«— 64 
4«4-i6  )  —48^2 4-96^—64 

27«^— 54«^-^'  +  36«.^-^— 8.r3  (  2,a  —  2X,  Ans, 


2765^ 


2  7  «M  —  5  Aft'^x  +  3  6«:r2 — S.r^ 
1 8a2'  4-  4-2^^  —  5  4«''^  +  3  6«^ — 83^ 


226 


CUBE     EOOT. 


^ 

O 

OJ 

•^ 

•  I— 1 

P 

1^ 

<X> 

O 

-+J 

m 

a> 

•  f—t 

• 

O 

•  r— t 

P 

o 
Q 

•l-H 

•i—i 

f-H 

P 

rH 

H 

1 

rzi 

h-t 

• 

1— 1 

o 
o 

-- 

O 

g 

OJ 

^ 

•  I— 1 

i-H 
O 

CO 

VO 

•  I— 1 

o 

1 

• 

P^ 

1 

OD 

CQ 

a 

^3 
•  1— 1 

1— 1 

H 

+ 

'r-t 
•  r— * 

CO 

+ 

CO 

1— 

+ 

M 

(N 

a 

"^ 

e 

ro 

ro 

ro 

fO 

+ 


CO 

52 

O 

00 

M 

1 

I 

Ttf 

■>!)< 

e 

C 

lO 

M 

M 

M 

+ 

+ 

in 

in 

« 

SS 

VO 

VO 

+  + 


VO     VO 


+    + 


;h 

^ 

0 

0 

m 

OQ 

•  r-H 

•  (— ( 

• 

>■ 

k 

?-i 

•  r—i 

•   f— 1 

?; 

O 

02 

P 

P 

O 

CO 

•  1— ( 

•  l-H 

o 

•  r— 1 
P 

OH 

•  1—1 

P 

•f-H 

P 

r— H 

s 

0 

•  1—1 

B 

0 
0 

• 

•  1— 1 

CD 

1— 1 

O 

o 

Q 

1 

CO 

+ 
CO 

^ 

P 

O 

CO 

1 

"^ 

+ 

1 

CO 

1— 1 

5^ 

« 

•\ 

o3 

VO 

Ov 

HH 

-* 

5^ 

CO 

+ 

4- 

+ 

■    Eh 

« 

ro 

rO 

^x 

f 

^ 

+ 

+ 

-- 

CO 

CO 

1 

PR 

•?. 

"^ 

■"^ 

in 

in 

1 

CO 

VO 

VO 

^ 

H 

1 

1 

1 

VO 

VO 
1 

1 

<9 

<D 

«5 

CO 

«o 

1 

0 

CO 

« 

&^ 

^ 

5^ 

C^ 

^ 

1^ 

ro 

rO 

ro 

ro 

CO 

CO 

CO 


M 

+ 

»— 1 
+ 

l-H 

+ 

to 

CO 

CO 

CO 

CO 

1 

1 

00 

00 

1^ 
00 

"^ 

1^ 

ON 

CO 

■o\ 

+ 

+ 

+ 

-- 

+ 

VO 
1 

1^ 

1 

>2 

1^ 
CO 

CO 

1^ 
CO 

1 

1 

VO 
1 

1 

VO 

VO 

1^ 
VO 

1 

VO 

1 

00 

1^ 

00 

1^ 

On 

1^ 

VO 

1^ 
CO 

1^ 
CO 

+ 

-- 

— 

0 

CO 

CO 

CO 

+ 

1 

T 

1 

to 

CO 

CO 

1 

1 

1 

. 

I 

factori:n-g.  22? 


FACTORING. 

1.  x^  —  gx  -\-  20  :=  {x  —  ^)  {x  —  5),  Ans. 

Extract  the  sciuarc  root  of  tlie  first  term  for  the  first 
term  of  the  factors ;  resolve  tlie  third  term  into  two  factors, 
whose  sum  is  tlie  coefficient  of  the  second  term,  and  thes:/ 
will  be  the  second  terms  of  the  re(|uired  factors.     Thu» 

^/x^  —  X',  20  =  {—  s)  X  (—  4) ;  —  5  —  4  =  —  9- 

2.  a^  -|-  7^  —  18  =  («  +  9)  {a  —  2),  Ans. 
's/ifi  =z  a\  —  i8=— 2  X9;  9  —  2  =  7. 

3.  a^  —  \T,a  -{-  40  =  {a  —  8)  («  —  5),  Ans. 
^^=a',  40  =  (-8)(-5);  -8-5  =  -13. 

^.  2al)c^  —  i^adc  —  ()oab  =:  2ad  {c^  —  7c  —  30) 

=  2al)  (c  —  10)  {c  +  3),  A71S. 

^Jc^  —  C  —  30=—  10x3;   —  10  +  3  =  —  7- 

5.  xHp  —  2XIJ  +  I  =  {xij  —  i)  {xi/  —  i),  Ans. 
\/xhf  —  xy;  i  =  (—  i)  (—  i) ;  —  i  —  i  =  —  2. 

6.  2>x^  —  32?/2  =  8  (.r^  —  ^if)     (Art.  131) 

=  8  (.1-  +  2?/)  (.T  —  2y),  Ans, 

7.  a;2  +  y'^z  4-  2/»?2  =  ^"^  +  ^  (i/  +  '^0  2'  -^^^' 

Note. — It  sometimes  occurs,  as  in  this  example,  that  only  a  portion 
of  a  polynomial  can  be  factored  when  there  is  no  factor  common  to  all 
the  terms. 

8.  i2a^x  —  Mhj  +  4^2  =  4«  (3^^^  —  2^y  +  2),  Ans. 

9.  Extract  the  cube  root  and  thus  find  one  of  the  three 
equal  factors : 

{a  —  x)     (a  —  x)     (a  —  x),  Ans, 

10.  1  —  a^  =  {i  +  d^)  (i  +  a)  (i  —  a),  Ans.  (Art.  131.) 


228 


G.    C.    D.      OF     POLYXOMIALS. 


II.   I  +  2rt  )  I  +  8«3  (  I  _  2«  +  4«2        (Art.  132.) 

I  4-  2a 

—  za 

—  2a  —  40^ 


4^2  -f  2>a^ 

Hence  (i  +  2c«)  (i  —  2rt  +  /^cfi),  Ans. 
12.     «6  _  ^4^.2  _  (^3  +  §2^^.)  (^3  _  ^2^^.),  ^^^5.    (Art.  131.) 


G.   C.    D.    OF    POLYNOMIALS 


l*(i*je 

'><SN^. 

I.                   4^2  _ 

^ax—isx'^ 

6^2  _j_yf^;^'_  3.^2     I st  Divisor. 

Multiply  by  3 

2                           I  st  Quotient. 

ist  Div.       \2a^— 

i2ax—4^x^ 

i2r^^  + 

i4ra'—  6.1^ 

6f^2  _j_  yaiv—^x^     2d  Di\id. 

Cancel.,  —i^x)  — 

26ax — 39.t2 

6^/2  4-  gax 

2d  Divisor 

2rH-3-^' 

—  2ax—3X^ 

2d  Quotient 

3«— .T 

—  2  ax — $x^ 

2a  +  3^,  ^^5. 

2.      4^^;2;^,    ^4?25. 

3.      16:^2  —  .v2  : 

=  (4-^'  +  //) 

(4^'  -  y) ; 

1 6.^2  —  8.?;^  -f-  ?/2  =  (^x  —  ?/Y; 
Hence   4.T  —  y,  Ans. 


4.  I  st  Di  V.  =  6^2  _|_  I J  fjj^  _|_  ^^2 

6«2-|_    7^2' — sx^ 

Canceling.  2X  )  4ax  -f-  6/- 

2d  Divisor        r=        2r/  +  3.?- 
2d  Quotient     =      3a— x 


6^2  _|.  y  ax — 32-2  =  I  st  Divisor. 
I  =  I  st  Quotient. 

6a'^-\-  7fl'a;—3.i'2=  2d  Dividend. 
6a^  4-  gax 

—  2a  X — ^x^ 

—  2  ax  —  ^x^ 


2«  +  3.r,  Ans. 


L  .    C  .    M  .     OF     P  0  L  Y  K"  0  M  I  A  L  S .  229 

5.  a^-¥  =  {li^  +  y^)  (a  +  h)  {a  -  h) ;     (Art.  131.) 
a-  —  h\i^  =  a^  ((v^  —  y^)  =  a^  {a  +  I)  {a  —  h). 
Herce  {a  +  Z/)  {a  —  h)  =  a^  —  ¥,  Aus. 

(Art.  140,  Prill,  i.) 

6.  a.-3  —  «3  —  (.^.  _  rt)  (ic2  _^  ax  +  rt^) ;     (Art.  130.) 
xi  —  a'^  =  (a-2  +  «2)  (^x  +  a)(x  —  rt);     (Ai't.  131.) 
Hence  x  —  a,   A71S. 


L.    C.    M.    OF    POLYNOMIALS. 

raf/e  284, 

I.     6a^  —  4a  =  2a  {yt  —  2),  prime  factors; 

4a^  +   2rt   =   2rt  (2«2  +    i), 

6(1^  -{-  4a  =^  2a  {yi  -\-  2), 

2a{2a^-\-i)(yi-\-2)[T,a  —  2)  =  $6a^-\-2a^—Sa,  A)is* 


2.  4(1  +«2)  ^  22(1  -^a^); 

4(1    _,,3)    ^    22(1    ^_^)(i    _«). 

8(1—  a)  =  2^  (i  —  «) ;     8(1+  rO  =  23  (i  +  n'). 
Hence  2^  (i  4-  f^2)  (^i  _^  a)  {i  —  a)  z=  8(1— «*),  .i;i.«. 

3.  «3  —  2rt  +  I  =z  (r^  —  i)2; 

a^  —  1  =  (ft2  4.  i)  (,,  _^  i)  (^^  _  i)  . 

a^  +  2ft  +  I  =  (ft  +  i)2. 
Hence  (ft+i)^x  (ft— i)2x  (ft2+i)=  (ft^  —  1)2  x  (ft^  +  i) 

=    (ft2-    I)   (ft4_    i) 

=  a^  —  c&  —  ft2  4_  i^  j;i^. 

4.  12  (ft&2  _  (^3^  =3  2^  X  3^2  (^  _  J) . 

4  (ft'^  +  ftl^)    =    2%  (^  +  J)  ; 

18  (ft2  _  /;2)    ^    2    X    32  (ft  +   ^)  (^/  -  l\ 

Hence  22x  32ft^2(f^^^)(^^_j)_36f^^(^2_^2)^  ^^^^^ 

5.  4ft2  —   I    =:    (2ft   +    l)  (2ft  —   l). 

The  other  two  quantities  are  prime. 

Hence  (2ft  +  i)(2ft  —  i)(4ft2  +  i)  =  i6ft^  —  i,  Ans, 

6.     See  example  2. 


230 


FRACTIONS. 


FRACTIONS. 
ra(/e  2S5. 


5^  + 

4X  — 


X 


3 

3-^—3 


Sx 


5^  + 
4X  — 


gx  —  9 


Adding, 


<X^  —  IQX  4-  q        . 
gx  -f ^~ -,  A  us. 


15^ 


a^ 


+  3i ta''>     (Art.  ioi.) 


Z»2   '   a^  —  h^ 


a?  +  W  +  2ah        (a-\-hf 


a  +  l) 


3'  3  +  5  +  7 
15 
15 


a  —  b       b  — 
ab  be 


■   ^4_^4        a^_a^j^ab^—b^ 

xa  —  2X       2a       x  —  a 

^ -j I • 

X  X  a      ' 

3«^  —  2ax  —  2rt^  —  x^  -\-  ax 


Ans. 


ax 


a 


2 


tt*^       ■■  ■     •v 


ax 


,  Ans. 


c       c 

-  +- 


a  -\-  ac 


ac 


ac  —  be  +  ab  —  ac  -\-  be  —  ab  -\-  abc       abc 


And 

From 
Take 


abc 

abc  . 

-zr-  =  I,  Ans. 
abe 


abc^ 


a  -\-  zh  — 
3ff  —    h  + 


d-h 

2 
d  -^  b 


—  2  a  -\-  4/1  -\- 


b-sd 


Or 


—  12a  -{-  24h  -\-  b  —  ^d 


,  Ans. 


8. 


FRACTIONS.  231 


X 

6.     From  $x  -f-  t 


Take  2X  — 


h 
x-h 


3'-^  +  A  + 


X      X  —  h 


h    '        c      ' 

^                                  ex  -\-  bx  —  ¥      . 
Or  3^  + f^ '  ^^^^' 


X 

7.     From  a  -\-  X  -\- 


x^  —  y^ 


Take  a  —  x  -\- 


X  +  y 


2X  4- 


X  X  —  y 


x^  —  y^      x^  —  y^ 


J 


Or  2X  -{-  -T-^— T,,  A^is. 

x^  —  y^ 

Multiply  the  terms  of  the  2d  fraction  by  x  —  y. 

a  —  I  a 


Or 


I  —  a; 

1  -\-  X 


a 

a  —  I 

i 

^2 

—  2(1 

+  I  - 

a^ 

«2. 

—  a 

I    - 

—   2ft 

A71S. 

a2 

a  ' 

I 

T 

+ 

X 
^2  ' 

(Art. 

183.) 

X 

I 

I 

x^ 

y 

I 

X 

I  —  X^        I  —  x^ 

r.       X  —   I  1   —  X  1  . 

Or, ■„  =  —  7 — ^-7 r  = ,  Ans. 

I  —  x^  {1  +  x)  {1  —  X)  I  -\-  X 


2o2  MULTIPLICATION     OF     FRACTIONS. 

MULTIPLICATION     OF     FRACTIONS. 
a  ~h       a  -\-  b  —  a  -j-  b  2b 


I.     I 


2  + 


a  -{-  b  ~  a  -\-  b  ~  a  -\-  b^ 

lb     2a  —  2b  -\-  2b 2a 

a  —  b  a  —  b         ~  a  —  b'* 


2b  2a  Aab  , 

X ,    —  -~ To?    ^>is. 


a  -\-b      a  —  b       if  —  b^ 

i"  +  ^ 
^x      20 

2b  T^X 

ZX       4a 


Sab 
gx^ 


Sab  QX'^      , 

gx^  Sab 


7'2  /lOM/  _i_  /J/2 


X'^  —  2X1/  -\-  ir       X  -\   n         ,  .  ,  . 

I  ^  —  y  J I  \    ^  iji 

=  x-  —  \f,  Ans. 
Cancel  x  —  ij  from  iinmerator  and  denominator. 


7&  2a^  —  4«^ 

2«2  —  Sa^  21b 


'jb  2C^  (l  —  20) 


2^2  ( I    -^   20)  (l   —  2r/)  2  1^*- 

Cancelino'  common  factors,  etc.,  we  have 7-,  Ans, 

3  -f  6« 

_^_  X  ^^  ~  ^^  r=      ^^     X  (a-4-y)(^--y) 
^  +  ?/  ab  X  +  //  «^ 

a  {x  —  y)       . 
=  — ^ — ^  ,  Ans. 
0 


D  I  ^'  I  S  I  O  N     OF     FRACTIONS. 


'2'^3 


I  «— I  I  a  —  I 

=  5b(a  +  i),  Ans. 

i5^<  -  30  ^  3«^_  _  3  (5^  —  10)  ^       3^^ 

2«  5«  —   10  2«  5a  —  10 

=  — ,  Ans. 


8.     ^ 


a;y  xy 


X  +  // 


x-y 


^y  +  y^  —  ^y  ^  ^ji  -y'^  +  ^y 


X  +  y 


X 


y 


-~-  X  -^ —  =  — I f- ,  Ans, 

X  '\-  y         X  —  y  x^  —  y^ 


I.     n  + 


a  — 


DIVISION     OF     FRACTIONS. 

Page  285. 

2a  «'  —  3«  +  2«  _  ^^^  —  ^  . 


rt  + 


(f       3 

a  —  z              «^  —  3' 

2«                rt^  . 

—  3«  —  2^        (v^  —  5^  ^ 

n       3 

^  —  3               «  -  3    ' 

2« 

2^           ^^■■^  — «        <z  - 
—                 X 

-3 

.      Li     — 

a      3 

«  —  3         ^  —  3       ^^"^  - 
a  —  I 

-5« 

«^—  5 


,  A)is. 


I      I       ?/^  + 1  ,1      y^  —  y  +  ^ 

X      xy'^  xy^     ^     ^  y  ^  ' 


y 


II  I     //^  +  I 

X      xy^       '^  y  xy^         y  —  y  ^  i 


y  + 


xy^ 


,  Ans. 


Note,     ?/3  +  i  ==  (|/  +  i)(/  —  jV  +  i)-    (-^^t  129,  Prin.  3.) 


234 


DIVISION     OF     FKACTIOI^^S, 


I 

I  +--^ 

X 

I 

X  -\-  I     ^    x-  —  1 

~       X        '        x^ 

z 

X  -\-  I              x^ 

X 

t' 

^^i5. 

X              X^  —  I          X 

2>y     . 

2y  —  2' 

2y 

sy      X  ^     ^ 

2  {y       i)          2y 

3 

■4' 

^nS. 

ah  +  ¥ 

9 

h 

a  —  J) 

b  (a  H-  b) 

v^ 

a  —  b 

a^  —  ¥  ' 

(a       b)  («2  Jr  ab  +  b^)  '^ 

b 

a  +  b 
a^  +  ab-\r^' 

- 

I 

a 

1  —  a  -\-  a  -\-  a^        i 

+  «2 

^2 

I  +  a       ] 

[  —  a  ~ 

1  —  a^                I 

J 

ii>.+ 

a     \ 

(i  +  a^f        I  +  a^ 
'   (i       a^Y        I       «2 

^.2)2 

1  —  aJ 

+  a^f 

I        ^2 

^/iS 

• 

l^rtflre  286. 

3« 

2«  —  2 

sa 

2a  —  2 

X   —   —  -,   A71S, 

2a          4 

2« 

a  —  I 


X 


8. 


3  _  3^  + a:   .   4^  +  //  _  4^  ^    4  _  i6.r 

yj^y_  3  4  3        5^       i5.y 


a« 


x^  —  «/^ 


—  I 


x^ 


^  -\-  y    .  .?/   +  a?2  —  y^ 


y 


x^  —  ?/2 


5+  I 


.T^  —  ?/2 


rc2 


r 


y 

x^  —  y^ 


X 


.r^  —  ?y2  y 


J^ 


10. 


EQUATIONS.  235 


I  I 

a      ab^  If  -\-  \       b-  —  b  -\-  I 


I  ab^      '  b 


(&+i)(^-^+  I)  ^  h 


a^2 


,  Ans, 


EQUATIONS. 
Piuje  286. 

1.  Let  X  =z  price  of  house. 

Then     $850  —  x=     "      «  biirn. 

And  5a:  =  12  X  $850  —  i2.f 

Transposing,  17a;  =  12  x  $850 

Dividing,  a;  =  12  x  S50  =  $600,  house  ; 

And  $850  —  x  —  $250,  barn, 

2.  Let  12^  =  No.  of  C's  acres, 
Then  8./;  =       ''       A's     " 
And  gx  =       "       B's     '" 
Then  29^'  =145 
Dividing,  x  =  5 

8.1'  =  40  acres,  A's ;  ) 
gx  :=  45      "      B's  ;  V  Ans. 
i2:r  =  60      '•       C's,    ) 


Ans. 


3.     Let  6x  =  No.  liters  cask  holds, 

Then  2X  =   '•        '*^     in  it  before  leakage. 

X  =z     21  liters, 
6.C  =  126     ••      A  us, 


230  EQUATION' S. 

4.  Given       XX ~  —  4  =  ~ — — 

'^  4  3  12 

Clearing  of  fractions, 

36rr  —  3:^  +  12  —  48  =  2o:r  +  56  —  1 
Uniting  terms,  1 32:  =:  91 

ic  =  7,  ^?Z5. 

5.  Given  3^jL9^7^±i_ii±4^+6 
^  235 

Clearing  of  fractions, 

45-^'  +  135  =  70^'  +  50  —  96  —  24a;  H-  180 
Uniting  terms,      ;r  =  i,  ^^^5. 

^.  X  ■\-  %      X  —  6 

6.  Given — — y  x  ^^  x  -{-  2 

4  3 

Cancel  .f  and  clear  of  fractions, 

3.1'  4-  24  —  /^x  +  24  =  24 

Uniting  terms,  a;  =  24,  Ans. 

^ .  a:  +  8       ^;  —  6 

7.  Given  X  —  2  =  .t  H 

4  3 

Cancel  ^  and  clear  of  fractions, 

—  24  =  3.r  +  24  —  4:^  +  24 

Uniting  terms,      ^  =  72,  ^;i5. 

8.  Given     2X  -\-  \       (—  x 

"'  X 


(-^) 


3  ^        4 

Clearing  of  fractions, 

8a;  +  4  +  3.1'  -f  9  =  I2X 

Uniting  terms,  ^  =  13,  yl';i6'. 

9.     2^  =  If  X  7  =  ff  =  7^3^X1  o'c'^^<^^'^  P-^i-  -1'^*^ 
^ce  Art.  247, 


lO. 


EQL'ATIOyS.  23T 

Let  27.T  +  £200  r=  stock. 

50  =:  annual  expense. 

2^x  +  £150 

gx  +      50  =  gain  ist  year. 


a        a 


36.T  +  £200  =  snm 

50  r=  expense. 

36:6-  +  £150 

12a;  +       50  =  gain. 

48^;  +  £200  =  snm  2d  year. 

6^x  +  £200  =    "     3d     " 
By  conditions,  64.^  +  £200  =  54,>-  +  £400 
Transposing,  lo.r  :=  £200 

X  =  £20 
And  273^  +  £200  =  £740,  Ans. 

rage  287. 

I.     Given  ^^^ — - — -  =  26 

3 

,     -                              6a;       6?/ 
And  ^ 


'  }•  Ans, 
=  9.   ) 


(I) 


=  0  (2) 

23 

From  (i),  4rr  4-  6^  =  78  (3) 

"      (2),  9a;  =  6//  (4) 

Substituting  in  (3),     4.'?;  +  9:^  =  78 

.*.    .«•  =  6 
And  from  (4),  y 


2.     Given  -  +  -^  =  8  (i) 

32  ^  ^ 

And  -  -{-^  —  -]  (2) 

23 

Multiply  (i)  by  18,     6x  +  9//  =144  (3) 

"        (2)byT2,     6x  +  4;/  —    84  (4) 

Subtracting,  57/  =    60 

?/  =     12  ;  ) 
Substituting  in  (4),  x  ^=z      6.   i 


238 


£Qi 

'AT  I  0:^5*^. 

Given 

ZX       ^  _ 
42^ 

(I) 

And 

2X  —  211 

4    '^      ' 

u) 

Multiply  each  by  4, 

SX           21J   —   36 

(3) 

And 

2X  —  2y         16 

(4) 

Subtracting, 

'  >  Ans. 
y  -  12,  \ 

Substituting  in  (4), 

Given 

2       3 

(I) 

And 

X       y 
3       4 

(^) 

Multiply  (2)  by  36, 

\2X  -\-   9?/            Tfic 

"        (i)  by  24, 

\2X  -\-  Zy        24b 

Subtracting, 
Substituting  in  (i), 

y  _  36r  —  24^  ;  [ 
ic        181^  —  24c,    \ 

^W5. 

Let                         X 

Price  paid  for  harness, 

Then                    dfic 

..       ..     ..   i^iorse, 

^ 

And                      3^ 

-       "     ''   buggy. 

Hence,                  ^x 

$400 

ic  =    I50,  harness  ;  J 
4X  =  $200,  horse  ;     ^  Ans. 
ZX  =  I150,  buggy,     ) 

6.     Let  lo.'T  +  ?/  =  The  Number. 

Then       loa;  +  3/  +  (loy  +  x)  =  121  (i) 

And         loa;  +  ?/  —  (lo?/  -^  x)  ^      9  (2) 
From  (i),                   11.7;  +  ii?/  =  121 

Or  a;  +  ?/  =     II  (3) 

From  (2),  X  —  y  ^       i  (4) 

Combining  (3)  and  (4),  x  z=  6 

And  y  =  5 

Hence,  loa:;  +  ?/  =  65,  Ans. 


E  Q  U  A  T  I  0  X  S  .  230 

7.     Given                         x  -^  y  —  z  =  o  (1) 

And                            X  -\-  z  —  y  ^=  2  (2) 

And                            y  +  ^  —  a:  =  4  (3) 

Adding,                     x  -\-  y  -\-  z  =  6  (4) 
Subtracting  (3)  from  (4),       2X  =  2 


(2)     "      (4), 
(I)     "      (4), 


And 


Ans. 


8.     Given  f +  -  =  ^  (i) 

And  I  +  I  =  ^  <^) 

And  ^  +  ^  =r  I  (3) 


.  -.  T  2a;       2?/       22  /  X 

Adding,  T'^   r~^'d  "^  ^  ^^^ 

2  ?/       22 
Subtracting  2  times  (3),       -^  +  —  =  2 


2.7; 

211 

Subtracting  2  times  (2)  from  (4),  ^^  =  i 

0 


cc 


2        "        (I)        -        (4),     -I   =    I 


And 


A71S. 


(C 
(C 
(C 


A  ns. 


240  EQUATIONS. 

9.     Given  iv  -\-  x  -\-  y  ^=:  6  (i) 

w  -^  X  ^  z  —  ^  (2) 

w  +  ^  +  .  =  8  (3) 

'^  +  Z/    f   :^  =  7  (4j 

Adding,      3^6;  +  3^  +  3.^  +  3^  =  3° 

Or  w  -\-  v  -\-  f/  -{-  z  =  10  (5) 

Subtracting  (4)  from  (5),         iv  =:  3 

(3)     "     (5),  ^  =  2 

(2)    "   (5),       y  =  I 

(i)     -     (5),  z  =  4 

10.     Griven  xy  =  600,     or    y  =  — -  (i) 

300  /      X 

a^z  =  300,     or     z  =z (2) 

yz  =  200  (3) 

Substituting  in  (3)  the  values  of  y  and  z, 

600  -200 

?/5;  = X  - —  =  200 

Reducing,  x^  =  900 

Extracting  root,  *   ic.  =  30  ) 

From  (i),  ?/  =  20  >■  A71S. 

From  (2),  2;  =  10  ) 

ir.     Given  ic  -f  •-^- =:  10         (i) 

48  ^  '^ 

X       y       z 

-  +  -  +  -==  22  2 

234 

4  +^+-  =  33  (3) 

Multiplying  (i)  by  16,    i6x  ^  ^y  —  2Z  —  304       (4) 

(2)  ''    12,        6X  +  4?/  4-  32  =:    264  (5) 

(3)  ''     A,        X  -\-  4y  +  2Z  —  132        (6) 
Subtracting  (5)  from  (4),         lo.^;  —  52  =  40         (7) 

(6)     "     (5).  5-^  +  ^  =  132        (8) 

Multiplying  (8)  by  (5),  25.T  -{-  s^  =  660       (9) 

Add  (7)  and  (9),        35;?;  =  700     .-.     x  —  20  ) 
From  (8),  2  =  32  >  Ans. 

(3).  y  ^  12^  . 


a 


EQUATIONS.  241 


12. 


Given        w  -\-    50        x, 

01-       .1: 

—   lU 

50       (l) 

:C  +    120    =:   3^', 

Ol'  3i/ 

—    X 

120        (2) 

y  -\-  i  20  —  2z, 

or    24; 

—  11  — 

120        (3j 

^  +  195  =  3^'-'r 

or  7,1'' 

<J    — 

195        (4) 

Add  (i)  and  (2), 

sy 

—  w 

170        (5) 

(4)  X  2  +  is), 

6/c 

y  - 

510        (6) 

(6)  X  3  +  (5), 

iSw 

—   ^6' 

1530  +  170 

Keducing, 

I  J  IV 

1700 

• 

.     w 

100  V 

^    '  Ans. 
90/ 

Substituting  value  of  ic  in 
"                "       ?t'  in 

(5), 

X 

y  - 

"                "       w  in 

(4), 

z 

105 

13.  Let  .T,  ?/,  and  z  rej^resent  the  ages  of  A,  B,  and  C. 

Then                                   x  +  sy  ^  3^  =  41^  (i) 

And                            '       4X  -{-    y  -\-  4Z  =z  580  (2) 

And                                   s^  +-  5^  +    2  =  630  (3) 

Add  (i)  and  (2),             ^x  -\-  4y  -{-  jz  =:  1050  (4) 

Subtracting  (3)  from  (4),      —  y  ^  6z  —  420  (5) 

Multiplying  (i)  by  4,      4X-{-i2ij-]-i2Z—  1880  (6) 

Subtracting  (2)  from  (6),       11//  +  8^  =  1300  (7) 

Multiplying  (5)  ])y  11,     —  11^  +  66?  =  4620  (8) 
Add  (7)  and  (8),                                745;  —  5920 

.-.     z  z=  So  yrs.,  O's  age  ;  i 
From  (5),  y  =  60  yrs.,   B's  age  ;  -  Ans, 

From  (i),  X  =z  ^o  yrs.,  A's  age,   ) 

14.  Let  X,  y,  and  z  be  the  numbers. 

Then  x  4-  y  +  z  =  S9 

X  —  11 

And    ^-  =  5.  or  x  —  y          =  10  (i) 

And    ' =  9,  or  x          —  z  =  i2>  (2) 


Adding  and  reducing,  x  =  29,   ist ;  ) 

Substituting  in  (]),  //  =  19,   2d:  ^A 

(2),  Z  =    IT.    3d.     ^ 


ns. 


a 
11 


242  gekeralizatio:n". 

GENERALIZATION. 
raye  '^SS, 

en  5  X  6       30  ,  . 

1.  X  ^ =  ^, -—  —  =  lo  hours,  Ans. 

p  —  c        S  —  s         3 

General  Problem. — Given  two  objects  starting  at  dif- 
ferent times  and  moving  at  different  rates  of  speed,  to  find 
the  time  required  by  one  object  to  overtake  the  other. 

Rule. — Multiply  the  rate  of  the  first  by  the  number  of 
hours,  or  periods  of  time,  between  starting,  and  divide  the 
product  by  the  difference  between  the  rates. 

abc  .  2  X  5  X  10  ^  , 

2.  X  =  — —  = r=  i±  days,  Ans. 

ab  +  ac  +  be        10  +  20  +  50 

General  Problem. — Given  the  times  required  for  n 
forces  separately  to  produce  a  result,  to  find  the  time 
required  by  the  united  forces  to  produce  the  same  result. 

EuLE. — Divide  tlie  product  of  the  numbers  denoting 
the  time  required  by  each  force,  by  the  sum  of  the  differ- 
ent products  of  the  same  numbers  taken  n  —  i  in  a  set. 

^.     Formula     x  ^=^  -  \ 


an        s  X  4400        „  . , 

ax  =  --  = ^^^—  —  liooo,  A's; 

5  22 

^  b?l  7     X    4400  „  -r>,        (      . 

bx  =  —  — -^-^  =  I1400,  B's;  \Ans, 


CM        10x4400        dh  n^ 

ex  =z  z= =   $2000,    C  S, 

S  22 

General  Problem. — The  proportions  being  given,  to 
divide  a  number  into  proportional  parts. 

Rule. — Divide  the  nnmber  by  the  sum  of  the  propor- 
tions and  multiply  tlie  ((uotient  by  each  of  the  proportions 
in  turn;  the  several  products  are  the  ])arts  re([uired. 


GENEllALIZATION.  243 


4.     Formulas, 


„        am  (n  —  i)        9  x  9  (3  —  i) 

F  r=z  ^^- —  ^ — ^-^ =27  years; 

m—n  9—3  , . 

a{n-i)        9x2  ^^^^^♦ 

S  =  -^ =  ^—7 —  —  3  years. 

vi  —  n  6 

General  Problem. — F  is  now  m  times  older  than  S ; 
in  a  periods  F  will  be  n  times  older  than  S :  to  lind  the 
age  of  each. 

EuLE — I.  Divide  the  product  of  the  given  interval,  first 
multiple,  and  second  multiple  less  one,  by  the  difference 
between  the  multiples ;  the  quotient  is  the  age  of 
the  older. 

2.  Divide  the  product  of  the  given  interval  and  the 
second  multiple  less  one,  by  the  difference  between  the 
multiples  ;  the  quotient  is  the  age  of  the  younger. 


EXPANDING  POWERS  OF  BINOMIALS 

I.     Powers  of  2a,  Za^  +    4^^    +    2^     -f     i 

''  —Zh,  I     —    3^*     +    9^2    _  27^3 

Coefficients,  1331 

{2a  —  TfiY  =  8a^  —  2,^o?b  +  54«J^  —  27^^^  Ans. 

2.  Powers  of  ^x,  812^+   2^01^  +     gx^    +   30:    +    i 

"  "    21J,       I       +        2Z/      +        4?/2      +     8?/3     +162/ 

Coefficients,       i  4  6  4  i 


(3a:  +  2yy  =z  Six^-^2i6x^i/-{-2i6x^y^-{-()6xij^-\- i6ij^yA7is, 

3.     Powers  ofi,     1+T    +    1+      i      -f-i 

"        "  3^>    I  +    3«  +    9^^  +    27^3  4-  Sia'^ 
Coefficients,      146  41 

(i  +  3^^)^^  =  I  +  T2C/  +  54^2  _^  jo8a^  +  Sia^,  Ans. 


244  POWERS. 

4.  The  expoiiciits  decrer,se  and  iiicreuse  by  2. 
(x^  -j-  ij'^Y  =  -''^  +  3^^i/^  +  3'^"'^"^  +  l/^}  Alts. 

5.  Powers  of  a,        a^  -\-    c^    +    rt^    +  i 

Coefficients,         ^        2)  Z  ^ 

(a  +  cr^Y  =  «^  +  3«     +  3<^"~^  +  «"^  ^^5. 

6.  Powers  of  «2,      «^2_|_^io_p   ^8_|_   ^6_|_     ^^4_|_     ^^2_|_    j 

'^        "  — 2rt,    1  — 2rt  +  4rt^— 8r^3_[_  j6^4_22rt5^64«6 
Coefficients,         i.     d       15      20      15  6  i. 

(^2 — 201)^  =  rt^^ — i2a^i4-6ort^^ — i6ort^  +  24o«^ — ig2a'^-\-64a% 

Page  288. 

1.  «Z^:r'^  X  a^  ==  akf:^,  A71S. 

2.  a^^~%~^  X  a'Wx'^  =  a^'^x"^,  Ans. 

3.  ic-'"  X  .^~"  =  a;~'"-%  Ans. 

4.  2/"^  X  y^  ^  "if  or  I,  ^?^5. 

1.  6«~^  -^-  2>(^~^  —  2rt~3,  ^W5. 

2.  ^-^hc-^  -^  4aWc^  =  2cr^I)-^c-^\  Ans. 

4.     (a  +  x)-"  -=-(«  +  ^)-'"  =  ((1  +  x)",  Ans. 

rage  280. 

Transfer  by  changing  the  sign  of  the  exponent.    (Art.  279.) 

I.  ^y^  =  xhj^^         3.  ^^  =  3  X  2-^tv^.-i. 

z^  -^  2y~z 


MULTIPLICATION^     OF     RADICALS.  245 

UNITING     RADICALS. 
rufje  28U. 

1.  A/48  +  V29  +  V243  ==  4V3  +  Z^3  +  9  V3 

=  16 V  3,  -4/is. 

2.  V54^  —  a/96x"  +  \/24.i'  =  3^6.1-  —  ^\/Gx  -\-  2'\JGx 

=  ^/dx,  Ans, 

2'\/\b      —    2\/^b      =    ^\/2b   =       V^2^ 

3^2 J,  vl^is. 


4.     a;A/25a:2c  +  V36a:^c  =  ^x^Vc  +  6x^\/c 


5.  'v/Sof*^^-^  —  V^oa^x  =  4«2v'5.r  —  2a^^x,  Ans. 

6.  3V'i282;^^;2  —  4:?:'\/t6?/2  =  \2X'\/  2yz  —  %x^/  2yz 

=  ^xy  2ijz,  Ans.    . 


MULTIPLICATION     OF     RADICALS. 

rage  289. 


1.  (rt  +  2/)'^^  X  (Z*  +  //,)»^  =  a/(^  +  h)  {a  +  ?/),  ^^is. 

2.  4  +  2V2 

2  —    V  2 

8  +  4A/2 

—  41/2  —4 
8  —  4,  or  4,  ^7^s. 

3.  {x  +  2/)^  X  (x  +  ?/)^  =  (.1-  +  y)^  X  (.f  +  y)' 

—  ('^'  +  ^)^  ^ns, 


246 


REDUCTION     OF     RADICALS 


4.     2>^^/d  +  y  X  Wa  =  3b\/{cl  +  yy  x  4\^a'' 

=  i2b\/a"  (d  +  yY,  Ans, 


I. 


DIVISION     OF     RADICALS. 
raf/e  289. 


(a%)m  _^  («ic)»*  =  ( —  I  ,  A71S. 


2.     2/^x^ay  -T-  6\^a  =  4^Vy,  Ans. 


3.        \/l6(i^  —   I2a^iC  -^  2«   =    2«'\/4«  —  3a;  -^  205 

=1  Vvi  —  3X,  Ans. 
4.     {b  +  ?/)^  -^  (5  +  y)^  =  (b  -j-  y)K  Ans. 


5.     4aVcib  -^  2^ac  =  — -'a  /  —  =  2a\  / 

2  \   ac  V 


~Vbc,  Ans. 

0 


REDUCING     RADICALS     TO     RATIONAL 

QUANTITIES. 

rar/e  2S9. 

I.     2\/a  —  Vl,  Ans.  3.     A/3,   ^W5. 


See  Arts.  103,  317. 


4.  V5  +  V^j  -4^5. 

5.  4V2.T  -f  sVy?  ^i?^5. 


RADICAL     EQUATIOXS.  247 

6.     v^  —  '\/6  +  2,  Ans, 

This  factor  is  obtained  by  trial. 


Multiply 

V3   +    V2   +    1 

By 

\'  2  —  V6  +  2 

^/6  -h     2  +  V2 

—  a/6               —  3a/2  — 2A/3 

2  +  2  a/2  -f  2  A/3 

4 

We  multiply  first  by  A/2  to  rationalize  the  A/2  in  the 
given  denominator.  The  product  contains  two  radicals, 
a/6  and  A/2,  which  may  be  made  to  disappear  by  multiply- 
ing by  a  negative  quantity;  we  therefore  try  —  a/ 6,  and 
the  result,  2  —  2  A/2  —  2 A/3,  has  two  negative  radicals,  the 
same  as  in  the  denominator;  and  we  make  these  disappear 
by  multiplying  by  +  2. 


I. 


RADICAL     EQUATIONS. 

ratjv  2SU. 

Given 

^/x  +  I  —  a/i  I  +  X 

Squaring, 

X  +   2  \/x  +1             l\    -\-  X 

Reducing, 

Vx           5 

X       25,  Ans. 

2. 


Given  V^  +  18  —  A/5  =  Vx—'] 


Squaring,     x  +  18  —  iV'^x  +  90  +  5  =  .r  —  7 

Reducing,  \/$x  +  90  =  15 

Squaring,  5a;  +  90  =  225 

Transposing,  5:6'  =135 

/,       ;?;  =  27,  ^?zs. 


2^8  QUADRATICS. 

3.  Given,  ^/29■  —11=5 
Squaring,  x-  —  11=25 
Transposing,  :7;2  =  36 
Extracting  root,  2:  =  6,  AiiSo 

4.  Given  =  ^  ^3"+^ 

V  3  +  ^c 

Clear  of  fraction,  6  =  34-0; 

Transposing,  a;  =  3,  Ans. 

5.  Given  (13  +  V23  +  ^2)1  _  ^ 
Squaring,  13  +  V23  -f  ?/2  =  25 
Transposing,  A/23  +  //  =12 
Squaring,  23  +  ?/2  =  144 
Transposing,  ^"^  =  121 

y  ■=  11,  Ans, 


6.     Given  2\^  =  V^'  +  3« 

Squaring,  ^a  =  :r  +  T,a 

Transposing,  x  =  rif,  Ans. 


QUADRATICS. 

Page  290. 

^ .                                     -zx  —  ^                  ^x  —  6 
I.     Given  ^x  —  ^ ^  =  2X  +  ^^ 

Transposing  and  clearing  of  fractions, 

6x2  _  i8r  —  6.T  +  6  =  ^x^  —  i5.r  -f  18 
Uniting  terms,  3.^^  —  9.r  =  12 

Reducing,  ^2  _  ^2.  —  4 

By  Art.  334,  x  =  i±  \U  +  i 


Reducing,  ^'  =  i  ± 


» » 


X  =.4  or  —  I,  Ans, 


QUADRATICS. 


249 


2. 


Given           

X 

100  —  9.T 
\x'            ^ 

Cleiiriiig-,   dOfX  - 

-    100    +    ()X            12X^ 

Traiispijsiug. 

12.^^  —  73.6'            —   100 

Divieling, 

^2           737.                    10  0 
X             'i2-^                    -12 

Second  nietliodj 

7.             731      a/          I00_i_5329 
1                             •*'             2  4    m     V              12     i^     5  7^ 

Reducing, 

/v             7  3      12  3 
•^             24   ^    24 

.'.     X        4  or  2-jl2^   ^^^^^• 

Given 

3  2 

24 

Mnltipl3'ing  by 

2,       X             ^                       i^ 

Second  method. 

-7-3                I     _1_     /»/               I          1          1 

X             4   ^     V            16    "T    IS" 

Extracting  root 

,                 X        ^/\,  Ans. 

Given 

V4X  4-2        4  —  Vx 
4  +  ^x             Vcc 

Clearing, 

2X  H-  2'\/x  16  —  :r 

Transposing, 

2\^X            16  —  3a; 

Squaring, 

4:c        256  —  96a;  +  9a;2 

Or                      ( 

^a:^  —  100.?;        —  256 

Dividing, 

^2             I  0  0  J.                       2  5  6 

Second  method, 

')■             50_j_a/          256      1      2500 
•^^              9     It    A'              9         1         81 

Eeducing, 

^             50    _i_    14 
^    —    "9'     1     "9^ 

.'.      X           7^    or   4,    ^726\ 

Let 

,r       greater  number. 

And 

y       less  number. 

Then 

.T  —  7/        12                   (i) 

And 

rc^  +  ^2  _   1424                  (2) 

Squaring  (i). 

X?  —  2Xij  -^  If  —  144                 (3) 

Subtracting  (3) 

from  (2),         2./V/        1280                (4) 

Add  (2)  and  (4),     x^  +  22:?/ 
Extract  root,  x 

Combining  with  (i),  x 

We  have 
And 


+  if  =  2704 

+  ^  =  52 
—  ?/  =  12 


X 


=  20;  ) 


250 


QUADKATICS. 


6. 


8. 


Denote  the  numbers  by  x  and  y. 
Then  x  -\-  y  =  6 

And  x^  4-  ?/3  =  72 

Dividing  (2)  by  (i),  x^  —  Q:y  +  //  =  12 
Squaring  (i),  x^-  -\-  2xy  4-  y'^  =  36 

Subtracting  (3)  from  (4),        yxy  =  24 
Or  xy  =:  S 

Subt.  (5)  from  (3),  a;^  —  2xy  -^  y^  =  4 
Extract  root,  a;  —  ^  =  2 

Combine  with  (i),  x  -{-  y  =^  6 

By  addition,  x  =z  4. 

By  subtraction,  y  =  ^ 

Denote  one  part  by  x. 

Then  the  other  part  =r  56  —  ic 


(•) 

(3) 
(4) 

(5) 


;[ 


Ans. 


By  condition, 

562;  —  x^  -—  640 

Changing  signs. 

,         x^  —  S6x        —  640 

By  2d  method, 

X        28^^  —  640  +  784 

Reducing, 

X       28  +  12 

And 

S6  —X  —  16,  ) 

Let 

X       B's  hourly  progress. 

Then 

a;  +  3  ^  A's 

And 

— -  —  B's  time  on  the  road. 

X 

And 

'5°     ^  81  _  A's         "          " 

x  +  3 

By  condition. 

150      1    25        150 

X  -\-  3         3            X 

Reducing, 

6            I        6 

^  +  3       3        ^' 

Clearing,        18a;  +  x^  +  3;?:  =  18a;  +  54 

Or,  x'  +  3.r  =  54 

By  2d  method,  x  =  —  J  ±  ^54  +  f 

Reducing,  x  =  —  |  +  -^^ 

.*.    X  =  6  miles,  B's  rate :  )    , 
And  a;  +  3  =  9      "     A's     "     f 


QUADRATICS.  251 

Let  X  =  greater  number, 

And  1/  :=  less  ii umber. 

Then  X  —  t/  =  6  ( ! ) 

And  2y^  -\-  4j  =  x^  (2) 

From  (1),  ^~'  —  ■^6  -f  1 2_^  -f  if 

Substituting,  2Xp  +  47  =  36  +  12^  +  ^2 

Transposing,  //^  —  1 2^  :=  —  1 1 
By  2d  method,  ij  —  d  ±_  V^Ti  +  36 

Eeducing,  y  =  n,  less, 

From  (i),  X  =1  '  ^ 


II,  less,         ) 
17,  greater,  f 


10.     Denote  the  length  and  l)readtli  by  x  and  y. 

Then  ic  +  ^  =  42  (i) 

And  2;//  =:  432  (2) 

Squaring  (i),      x^'  -{-  2xy  +  //-  =1764 
Multiplying  (2)  by  4,  4^?/  =1728 

Subtracting,        X'^  —  2xy  -\-  y^  =z  ;^6 
Extract  root,  x  —  y  =  6 

Combining  with  (i),        x  -{-  y  =  42 

By  addition,  x  ==24  ft.,  len!  ;  ^ 

By  subtraction,  y  z=  iS  ft.,  bre.,  f  ^ 


II.     Let  X  =  A's  age;' 

And  ?/  =  B's  age. 


T  20 

Then  xy  =  120,  or..r  = (i) 

And  (x  ~  s){y  +  2)  =  120 

Or        2;^  4-  2X  —  3//  —  6  =  120  (2) 

-n  •  240 

By  substi.,    120  +  -^ — 31/  =  T26 

Reducing,  ^2  _|_  2^  —  80 
By  2d  method,  //=:  —  !+  a/So  +  i 

Reducing,  ^  =    8  yrs.,  B's  age ;  )  ^^^^^ 

From  (i),  ^  —  15  yrs.,  A's  age,  f 


252 


QUADRATICS. 


1 2.     Given 

\^x^  +  V^^ 

6\/x 

Sffuaring, 
Dividing  by  x, 
Extracting  root, 

X^   -{-    2X^  -{-   X^ 

X^  -\-   2X^  -\-  X'^ 

X^  -j-  X 

X 

X 

—  36^' 

=:   6 

By  2d  Method, 
Reducing, 

-  4  +  V6  +  i 

1  4_   5 

2  =1=    2 

* 

,\       X 

2,  Ans. 

13.     Given 

X  +  V^  +  6 

—  2  +  ^Vx  +  6 

Transposing, 

X  —  2 

2Vx  +  6 

Squaring, 

x^  —  4^'  +  4 

—  4;r  4-  24 

Uniting  terms. 

x^       8x 

X 

20 

By  2d  method. 

4  --  V20  +  16 

Or 

X 

4  +  6 

«     •    .       */y 

—  10,  or  —  2,   J??5. 

14.     Let 

Then                  x 

X       No.  lbs.  pepper  for  Cio. 
+  60             "        ginger  for  £20. 

And 

—        Price 

X 

of  pepper  per  pound. 

Ari'rl                            — 

20 

O'l  n  nr£iv                 ^^ 

xxllU. 

X 

+  60 

gingei 

Bv  conditions,    80  x |-ioox 

^"^           6c 

^^  +  6o        ^5 

Reducing, 

i6o- 

400 

+  60        '3 

Clearing, 
Uniting  and  divi 

i6o;6'  +  9600  + 
ding,           .i2  4- 

X        — - 
X  -         ' 

400.?'   rr:    132--+  780.^ 

2  2  0  v              9  6  0  0 
"13-'^                 13 

By  2d  method, 
Reducing, 

I  I  0  _l_  /t/9600     1     I  2  I  00 
13       -^        13        1        "13^ 

[10     13  70 
13'          13 

2:  =  20 
10 


And  price  of  pepper  per  lb.     —  =  ]^  —  i£,  or  tos.; 


A71S 


<< 


(( 


gmger 


ii 


20 


2: +  60 


^(7  ^  i"^j  01'  5^*' 


COLLEGE     EXAMINATION     PROBLEMS.         253 


COLLEGE  EXAMINATION  PROBLEMS. 

Pafje  :>U1,  Art,  J^V. 


•  ('5-'-?)-(-^^); 


5^ 

5c  ex  —  a  -\-  0        5  [rx  —  a  -\-  u) 

2.  (a^  —  b^)  -^  (a  —  h)  =  a^  +  a%  +  al^  +  b^,  Ans. 
See  Art.  129,  Prin.  i. 

3.  Given  x  4-  ^^ ^  =  12 -^ 

2  3 

Clearing  of  fractions,  6x-\-gx — 15  =  72  —  4.T  +  8 
Uniting  terms,  19:^  =  95 

4.  Multiply  3^45  —  7V5  =  2  A/5 
By                ■  VTf  +  2V9f  =  17VJ 
And  lyVy  X  2V5  =  34,  ^4w.*?. 

l7i  i?!  17-2-  > 

5.  «^03  -^  a-iO^  =  ci^b^^,  Ans. 

Subtract  the  exponent  of  the  divisor  from  that  of 
the  dividend. 

6.  xy~'^  -^  a.%~3  —  x^y~^,  Ans. 

7.  Given  2>^^  +  ix  —  9  =  76 
Reducing,  .r"^  +  |:?!  :=  -^ 

By  2d  method,  x  =  —  J  +  \/ ^  +  J 

Reducing,  :r  —  _  i  _[-  j.fi 

.-.     a;  =  5,  or  —  5I,  Ans. 


I,- 

— 

7^ 

8 

/ 

x^ 

3"^ 

X 

— 

5 
4 

1 

3 

Vj 

+ 

I 

9^ 

X 

— 

I 
3 

7 

• 
•    • 

X 

1  : 

h    ^1'    - 

5 

^7 

Ans. 

254       coLLECxE    examinatio^nt    problems. 

8.     Given  ^.t?  - 

Transposing,  etc., 
By  2(1  method, 
Eeducing, 


Denote  the  numbers  by  x  and  y. 

Tlien  X  \  y  w  X  ^  y  '.  dt2          (i) 

And  '                   X  \  y  \  \  X  —  ^  :  6            (2) 

From  (i),  (Art.  378.)      42.C  =z  xy  -\-  y'^ 

From  (2),  6x  =  xy  —  y^              (3) 

Subtracting, 

Or  x  =  ^                     (4) 


36.^ 

2lf 

X 

18 

t 

3 

18 

y' 

I 

y 

18 

I 

Substituting  in  (3), 

Dividing  by  y^, 

Reducing,  y  =  24,  less  ;  )  ^^^^ 

From  (4),  X  =  32,  gr.,     f 


10.  S  =  — ^ ,  =  ^^  =  -  =  li    .4??.s'.       (Art.  435.) 

^       ^         I       T        3 

11.  {a  +  ^)i2  =  r/i^  +  i5fti4j  +  105^13^2^  etc.,  Ans. 

12.  (i  +  .T2)-i  =  I  _  -  -f  ^-  —  ^ ,  ^?z.s.  (Art.  270.) 

5        25        125 

^  =  — i; 

/t' I  T  S"  I  J  1  ^ 

w  X =  -i  X  — ^ — -  =r  -J  X  -J  =  2^; 

2  2 

w—  I       n  —  2         -         —4—2 

w  X X =  -h  X 

233 

3  V  IT      I  .T 


I. 


COLLEGE  EXAMINATION'   PROBLEMS. 

r<i(ji'  2iH,  Art.  54:3. 

p  +  2y       .r\  _^  i^_±jy ^     \  . 

\x  -^  y      yl  '  \     y         i^  +  yr 

xy  +  2y^  4-  ^'^  +  xy  _^  x^  +  s^^y  +  ^y^  —  xy  ^ 

{x  +  y)y        '         {X  +  y)y 
2xy  +  2//^  +  x-^        {x  +  y)y     _  ^    .^ 

X  +  y)  y  2xy  +  2y^  +  :7;^ 


2«2 


Given  re  +  ^a^  -\-  x^  =  -  ,— 

Vcr  -j-  x^ 

Clearing,       xVci'^  +  x^-{-a^  +  x^  —  m^ 

Transposing,  x^/ a^  +  a:^  =  ci^  —  x^ 

Squai-ing,  a^x^  -{-  x^  —  a"^  —  2(rj?'  +  x^ 

Uniting  terms,  3rt2i~  =  a^ 

Or  x^  =- 

3 


«     /- 


4.     Given 


o 

90  90  27 


X        X  -\-  1        X  -\-  2 
Then      i ox"^  +  30.T  +20  —  i  ox- — 2o.r  —  3.^•~ — 30;  =z  o 
Eeducing,  3:?;^  —  7  a:  =  20 

Dividing  by  3,  a;^  —  ^.t  =  ^  • 

Bj  2d  method,  -^  =  f  ±  V^  +  || 

Extracting  root,  ^  =  I  ±  V" 

/.     a;  =  4,  or  —  I  J,  ^/i5. 

5.  Given  Va;  —  i  =z  x  —  i 

Squaring,  x  —  i  —  x^  —  2X  -{-  i 

Transposing,  a?^  —  3a;  =  —  2 

By  2d  method,  ^  =  I  ±  V—  2+1 

Eeducing,  a;  =  |  ±  ^ 

.*.     a;  =  2,  or  i,  Ans. 

a  "^ 

6.  s  = =  7^-3  =f  X  ^=5i,  A?is.    See  Art.  435, 


25G    COLLEGE  EXAMINATI-ON  PEOBLEMS. 

7.     Given        x  +  ^/ x  :  x  —  ^/x  ::  3^/^  +  6  :  2^/2:^ 

By  Theorem  7,   2.T  :  ic  —  ^/x  ::  5  V;6'  +  6  :  2^^- 
By  Theorem  i,  /^x's/x  =.  ^xVx  -\-  x  —  6\/x 

Eeducing,  4X  =  5.T  +  Vx  —  6 

Or,  Vx  =  X  —  6  (i) 

Squaring,  x  =  x^  —  i2;i"  +  36      (2) 

Transj^osiiig,  x^ —  i^x  =  — 36 


By  2d  method,  x  =  -y-  +  V—  36  +  ij^ 

Or,  X  =  J/-  ±  2  =  9  or  4. 

It  may  be  observed  that  this  is  a  peculiar  case.  Both 
vaUies  of  x  satisfy  equation  (2)  ;  while  equation  (i)  is  only 
satisfied  with  the  second  value  4.  The  first  value  9,  only 
reaches  backward  to  equation  (2);  while  the  second  value 
4,  verifies  tlie  given  proportion.     Hence,  x  =  4,  Ans. 

8.     (a^  +  xy^  z=a^-{-  i2aPx  +  66a^^x^  +  22oa^^2^-\- etc.,  Aiis, 

See  Arts.  270,  271. 

n  —  I        12x11        ^, 
?i  =  12  ;     n  X =  — =  66; 

2  2 

n—i       n—2 
n  X X  =  66  X  -V-  =  220. 


9.  ^x^-f)-i  =  iL-Q 


A-^ 


~  ^iV  "^  4:^2  "^  32:^4  "^  1282-6'  ^^^'Z 

=  "1  +  ^  +  — -1  +  ——1-3.  ^tc,  .b^s. 

ips         4.T-         322:2  1282;^ 

?2  =  —  J  Arts.  270,  271. 

n  —  I             J       —  i  —  I 
W  X =  —  I  X  — 

2  2 

—  ?^  IS    —    32 


COLLEGE     EXAMfXATIOK      PROBLEMS.  257 


71—1         n  —  2   _     5  —  i  —  ^   _     5  3 

n   X X         -         —   T2    ><  ^  —   '3  2    ><    —  ^ 

23  3 

15 

—  —  T2¥- 


JPrff/e  :^.9i,  ^i^  54^. 

I.     24saW-i-I  =  {3a^+I)(Sla^^  —  2'Ja%^-\-ga^I^—Sa^+l) 
Sia^^—i  =  {ga^b-^+i)  (3a%-h  i)  (sa^—i).  (Art.  129.) 
Hence,     </.  c.  c?.  =  3«^^  +  i,  ^ws. 
Again,  the  /.  c,  in,  of  these  factors 
—  (^ga^]j2  +  i)  (3^2^  4_  i)  (3«2^  _  i)  {%ui%^  —  2^aW 

+  ()nW  —  z(i^^  +  i) 

—  ga'^lr  +  3^<^Z'  —  I,  Ans. 

^.  ..  6\/h      ,  2oc\/i>^ 

2.  DiMde  - — J—     by — = 

25Vrt^  2\ahy  a^ 

6h^  2och^ 

• 

Q  •  5 

25^^  2  1^3^ 

6  X  2ia^b^  ,,    16,3     .      . 

25    X    2  0ft5^Tc 

3.  Given 
Or 
And 
Squaring  (i),  y^  =z  ^x^  —  84.C  +  441 

Substituting  in  (2), 

2X^  4-  4a;2  _  84a;  +  441  =  153 

Reducing,  x"^  —  14.C  =  —  48 

By  2d  method,  x  —  7  ±  V—  48  H-  49 

.-.     X  =       8,  or  6  ;       ) 

>  Ans. 
And  from  (i),  y  —  —  ^,  (^r  —  9,  \ 


2X  —  y        21 

^             2.i'  —   2  I 

(I) 

2a;2  4-  ?/2  _  153 

(2.) 

258  COLLEGE      EXAMINATION      PROBLEMS. 

rage  292,  Art.  544, 

4.  Let  X  =z  No.  of  yards. 
Then                               —  =  price  per  yard. 

By  conditions,  —  =  — ; 1-  i 

-^  X  X  -\-  2 

Clearing,  iSoiC  4-  360  =  180a;  -\-  x^  +  2X 

Or  x^  +  2X  =  360 

By  2d  method,  x  =  —  i  ±  Vs^^i 

=  —  I  ±  19 
/.     a;  =  1 8  yards ;  \ 

And  ^  =  $5,  price, 

5.  (a  —  by^  =  aP  —  \20>^h  +  (i(ia^%^  —  2  2oa^h^^A9S<^^^^ 

—  ']g2CVh^  +924«^Z'^— 792rt^^'  +  495«4Z'^ 

—  22oa%'^  ^6(}a^^^—i2al)^^-^l^\  A?is. 

6.  4^2  _  gy2  —  (2flr  +  3^)  (2fl!  —  3?/),  Jw5.     (Art.  103.) 


«5  /a        .     /  a^        ,     /2«^        6fl^ 


=  ^^2,  Ans. 

8.     Given  ic  +  2?/  =     7  (i) 

And  2X  +  3//  =12  (2) 

Mult,  (i)  by  2,  2X  4-  4//  =  14  (3) 

Snbt.  (2)  from  (3),  ^  =  2  ;  )  ^^^^^^ 

Substituting  in  (i),  a;  =  3,    f 

^.  a;  —  ^       X  X  —  10 

10.     Given -\ —  =  12 

32  3 

Clearing  of  frac,  2ic  —  10  +  3.6'  =  72  —  2^  +  20 

Uniting  terms,  7:?:  =  102 

X  —  14^,  Ans, 


COLLEGE     EXA3[INATI0N     PHODLEMS.  259 


Page  292,  Art,  ^45. 

I.     From        T.x  A — 7         subtract        x  —  ' 

2h  c 

.                  X       X  —  a  cx-{-2h  ix  —  a) 

Ans.  2X  -\ — r  H ,  or  2X  + 


x^ 


2b  c     '  '  2bc 

I  —  y-  I  x    \ 

X  — ,-^T,  X  1 1  H ),     or 

X  +  x-  \  \  —  XI 


1+2/       ^  + 

(i  +a:)(i-.r)(i  +  ^)  (i  -  ij)  ^  \  -y     ^^^^ 
{\  -^  y)x{\  ^  x){\  —x)  X     ' 

c&  —  \d^h  +  ZaW  +  ^¥  {  a^  —  2ab  —  2b^,  Ans. 
a' 


2a^  —  2ab  )  —  4a^b 

—  4d^b  +  4aW 

2^2  —  ^ab  —  2b^)  —  4nW  +  Ub^  +  4^^ 

,.     From  2^/320  =  8^/5 

Take  3'V^4o  =  6^/5 

2V^5,  ^7i5. 


1  tI 


5.  a'^b^  ~  a^b"^  =  cr'^b^^,  Ans. 

6.  Given  ^*  +  4.^^  =  12 


By  2d  method,  x^  ■=^  —  2  ±  A''^i2  4-  4 

Eeducing,  .t^  =  —  2  ^fc  4  ^=  2,  or  —  6 

Extracting  root,  a:  =  ±  A/2,  or±\/  — 6,  J?kv. 

Given  x^  —  a;  V3  =  :z^  —  ^Vs 

Transposing-,     x^  —  (1  +  A^)-^'  =  — 


2 


By  2d  methcd. 


3         ■"  V  2  4 


260  COLLEGE     EXAMIN^ATION     PROBLEMS. 


Eeduciug,        *      x  —  ^—^ — -  ±_  i 


2  2 


8.     Denote  the  numbers  by  x  and  y. 

Then  x  -\-  y  ^=^  2a  (i) 

And  ^2  _|_  ^2  —  2^  (2) 

Squaring  (i), 

x^  +  2xy  +  «/2  =  4«^  (3) 

Subt.  (2)  from  (3),       2:?;?/  =  4^^ —  2^  (4) 

"  (4)  "    (2), 

a;^  —  2xy  +  ?/2  =  4^  —  4fl'2 
Extracting  root,        x  —  y^  ±.  2\/b  —  n^  (5) 

Combining  (i)  and  (5),    r*::  =  «  ±  V^  —  c^ 


And  y  =  a^^  ^h  —  c^. 


Ans. 


:  2  :  3  (i) 

:  2  :  5  (2) 

:  5  •  3 


9.     Denote  the  numbers  by  x  and  y. 
Then  x  —  y  :  x  -[-  y 

And  a;  —  ^  :     .t// 

From  (i),  (Th.  7)  2x  :  x  -\-  y 

Theorem  i,  6x  ^^  <^x  -\-  <^y 

Or  X  =1  sy  (3) 

From  (2),  (Th.  i)  5a;  —  5?/  =  2a;?/ 
Substituting  value  of  x,  2oy  :=  loy'^ 

Reducing,  ?/  =     2    ' 

From  (3),  '  X  =.  10, 

10.     r  =  Q"""'  =  (iA2)i  ^  ^87  =  3. 

See  Art.  407. 

Hence  the  ceries  2,  6,  18,  54,  162,  .4?is, 


'  '  Alls. 


COLLEGE     E  X  A  M  I  K^  A  T  I  0  X     PROBLEMS.       2G1 


il. __^.^  =    7)1  ix-  +  ^2)     2    zz:   —  I  I    -f         I       ; 

'y/x'  +  «2  X  \  J- J 

/I  «2  ^^/4 


(^  —  ^^,  +  7r~- ^7?  etc.),  Ans. 

\x       2  r^       8:?:^       48.1-7 '         /' 


See  Art.  270.     y^  =  —  ^ ; 


w  —  I  _        J        —  i  —  I  _       J  3  _  i 

^i  X       ^        _  —  2   X  ^  —  —2  X  — :f  —  %; 

w  —  I       n  —  2        -       —  I  —  2 

n  X X ==  f  X  — 

233 

—    3    V     5    —  15 


rage  292,  Art.  546. 

12.?'— 102  I2(.T^  — 16)  000^ 

1.     -f—  =  — ^7 r-  =  4X^  +  8x^-\-  16.T+  32,  Ans. 

3'"  — 0  30'  — 2) 

2.  Divide        cc^  H 7   by y  —  m 

„   ,     .        x^  ia  —  h)  +  x^       ah  —  yn  (a  —  t) 

Keducmsr,  — ^ S '• S 

a  —  0  a  —  0 

-P^.  .  x^((i  —  Z*)  +  ^3 

Dividme^,    -~ j- tt,  Ans. 

^     ab  —  m\a  —  b) 

^.  3a;  —  II        S^  —  S       91  —  IX 

3.  Given  21  + —  =  •     „    ^  4-  ^ -- 

16  8  2 

Clear,  of  frac.,    336  +  3.^ — 11  =  lo.r — 10  +  776  — 56:^ 
Uniting  terms,  49X  =  441,     x  =  9,  ^ ;?,-?. 

i      2      3      _  4  6  7  . 

4.  a-a^a^a  »  =  flrso^  ^«,9. 

i  +  l  +  i-f="  3o±4CH^5^8  ^  6  7^  exponcnt. 

^,.  11;       72—62: 

5.  Given  -^  — r—  =  2 

X  2X^ 

Clearing  of  frac,  t,ox  —  72  +  62'  =  ^x^ 
Redncing,  x^  —  gx  =  —  18 

By  2d  method,  x  =  ^  ±  V— 18+"^ 

.'.     X  =  ^  +  ^  —  6,  or  3,  Ans. 


262       COLLEGE     EXAMINATION     PROBLEMS. 


6.     See  Art.  394.     d  = 


I  —  a 


50  -  I 


=  7. 


m  +1         6  -f  I 
Hence  the  series, 

I,     8,     15,     22,     29,     36,     43,     50,  ^^5. 


/'ttr/e  ;^.9.j?,  Art.  546. 

7.  EuLE. — Divide  the  product  of  the  natural  numbers 
from  m  down  to  7n  —  w  +  i  inclusive,,  by  the  product 
of  the  natural  numbers  from  i  to  71  inclusive. 

-CI      '  ^       w  .  .(m  —  n-\-\)        8x7x6x5  . 

Formula,    G  — ^ = =  io,Ans. 

1  '  2  '  2i  '  '  '  '  n        1x2x3x4 

Note. — m,  denotes  the  whole  number  of  letters,  and  n  the  number 
of  letters  taken  in  a  set. 


8.     (,  _  J)-}  =  L  f,  _  *^ 


.1 


=  -    I  +  —  +  "^  +  -Vo,  etc. 

(«  _  J)  T  ^  -_.+  -_  4-  -A_  +  __^^^^  etc., 
«T       4^1       32«T       1280^"^ 

9.     Denote  the  smaller  by  x. 


) 


Ans, 


Tlien 

By  Theorem  i, 

Or 


10.     Given 
And 

Subtracting, 
Substituting, 


a; :  150  —  .t  : :  7  :  8 

Sx  —  1050  —  yx 

15.2-  z=:  1050 

.-.     X  =  70,  smaller  ; 

150  —  .T  =  80,  greater, 

5-'^  H-  2//  =  29 

—  X  -\-    21/   =z  —   I 

6x  ■=  30 

I^  Ans. 


A71S, 


COLLEGE     EXAMTNAflOX     PROBLEMS.       2G3 
ruf/e  V?.'>.V,  Art.  547. 

1.  Given       [-  I  z=  o 

2    +  //  2/  —   2 

loy  —  20 — lo  —  5// +  3// — 12  =  o 
Uniting  terms,  3?/  -f  5?/  ==  42 

Dividing,  //  +  |^  =     4 

By  2(1  method,  ?/=  —  6^±  Vi4  +  |f 

Keducing,  /y  =  —  f  ±  -¥" 

•••     y  =  3?  <^>i*  —  4f,  ^^'^^• 

2.  c!^  ^  rt^  --  rt  '^ ;    rt~i  -^  r<^  =  /^r^.     Sec  Art.  396. 

Hence,  a,  a^,  a~  ^  are  part  of  a  geometrical  series  of 
which  the  ratio  r  =  cr^,  Ans. 

3.  Let  20:  =  first  number, 
And  47/  =  second  number. 
Then  x  -i-  ^y  =  11                        ( i ) 
And  a;  +  3y  =  6./'  —  4y               (2) 
From  (2),  X  —  —                         (3) 

Substituting  in  (i),    ^^  +  3?/  =  11 
Or  22?/  =  55 

^y  =  5 

^  =  I 
From  (3),  a;  =  I  X  I 

.-.     4//  =1  10,  second  ;  )     . 
And  2.T  1=     7,  urst,        ) 

4.  Powers  of  2a, 

256^8+    128^?''   -f-     64'/^     ..      4rt~     +    2rt    4-      I 

Powers  of  —     , 
3 
h  y^  ¥        W         h^ 


3  9  729      21   - 

Coefficients, 

I  8  28       ..    28  8 


(-  -  -3)' 


loz^a'!)     \']()2n%^      112(1%^     i6ah''       ¥       . 
3  9  729        2187      6561 


264      COLLEGE     EXAMIN^ATIOK     PROBLEMS. 

5.  (p  -W  =  {a  +  1))  {(I  -  b) ; 
«2  —  2ab  +  />^  =  {a  —  hy. 
Hence  the  (j.  e.  d.  is  a  —  b,  Ans. 

6.  a^  —  x^  =(«  +  :?;)  {a  —  x)\ 
Hence  I.  c.  in.  is  4^^  —  42;^,  Ans, 

\7,a  —  2()b       7  J  —  21a       gh  —  iia 

i3«2_42fl',^  +  2  9^^ — 2'iab-\-2ia^-\-']b'^-]-()b  —  11a 


5  («  -  <^j2 
34f/2  —  63r/Z>  4-  36^^  +  gb  -^  iia 


,  or 


5  («  -  ^)' 

8.     Given  ^/x  -\-  a  =  Vx  +  ^ 

Squaring  x  -\-  a  =z  x  -\-  2a\/x  +  «2 

Eeducing,  2^/x  =:  i  —  a 

Squaring,  4.T  =  i  —  2a  -f-  a^ 

I  —  2«  +  «2 


Par/e  293,  Art.  54S. 

a^4-ah' — ax^ — x^       (a-\-xY(a — x) 

I.     9 — ^ =  ;— ^ — (-7 i  =  a+x,  Ans. 

a?—x?  [a-{-x)  [a — x) 


2.     ar%'^  X 


a-W  -^  -^— r  = —-,  or  -^ ,  or  «    3  h^, 

7ic  —  6  :r  —  ^  X 

Given  ■^- 7 ^—  =  - 

35  6x  —  loi        5 

Multiplying  by  35,  jx  -  6  -  j^^^  =  7^ 

Transposing,  etc.,  362: — 606  +  352' — 175  =  o 
Uniting  terms,  yix  =  781 

.-.     X  =  ii"  Atis. 


COLLEGE  EXAMINATION  PROBLEMS.    265 

n-  7^'  +  9        /  2./;  —  i\ 

4.  Given —  (x 1  —  7 

4  V  9      / 

Clearing  of  fractions,     63^  +  81  —  28.^—4  =  252 
Keducing,  35^  =  ^75 

Of        00 

5.  Given +  yf  =  8 

Trans,  and  mult,  by  2,  x^—  ^x  ^=  ^ 

By  2d  method,  x  =  I  ±  Vi  -\-  ^ 

Eeducing,  rr  =  |  ±  | 

.-.     X  =  i^,  or  — f,  A71S, 

6.  Denote  the  numbers  by  x,  y,  and  z. 

Then  xy  =  15 

Or  ^  =  'I  (i) 

And  xz  =  21 

21  /     X 

Or  -'^^  ==  V  (2) 

And  /  +  2;2  _  y^  (3) 


Equating  (i)  and  (2), 


15    _    21^ 

y  ~   z 

-^        21 

^       441 

Substituting  in  (3), \-  z^  —  74 

°  441 

Or  666z^  =  74  X  441 

.-.     z^  =:  4.g 

Extracting  root,  2;  1=  7,  3d  ;  ) 

Substituting  in  (2),  ic  =  3.  ist ;  )-  Ans, 

Substituting  in  (i),  y  =  5,  2d,   ) 


266        COLLEGE     EXAMIKATIOJS"     PROBLEMS. 

Page  2i)4,  Art.  548. 

7.     I  z=  a  -\-  {n  —  1)  d  z=z  I  -\-  (n  —  i)  x  i  =  ■?^; 

See  Art.  388. 

a-\-l  i-\-n  n-{-n^      . 

S  = xn  = xn  =  — ,  Ans. 

222 

See  Art.  389. 

«.    K -.,-'  =  ;■(.-!)-' 

=    -  (  I    +    — -3  H -6  +    -^9  +  -T2'    etc.) 


J3  2^  14^9  ^5^2 

30*  "^  96?^  "^  8i«io  "^  243^ 


I  +  t:::!  +  r::^  +  ^7::rn  +  7^3'  etc.,  Ans. 


n  =  —  J  ; 


?i  —  I            ,       —  i  —  I 

2  . 

^2-          —            ^                        2                             3     ><           3 

9  > 

n—1     n  —  2        _      — J — 2 

14  . 

/t-  X                  X                    —    g  A                          0  A          Q    — 

233 

^1  J 

n—1       71  —  2       /^  —  3 
^z  X X X 

3 

14     w  3^  3    —    I4v     5    —    _3_5 

—    —  ^T    ^  ]  —  ^T-^  ^   —    24  5' 

4 

9.     A/300  +  A/75  =  10 A/3  +  5a/3  =  15 V3,  ^^2^. 

10.     Given  -  -\ —  =  -7- 

7       ^  +  S 

Clearing  of  frac.,  x^  +  5^  +  147  =  23a:  +115 
Transposing,  x^  —  i8ic  =  —  32 


By  2d  method,  x  =  g  ±  a/— 32  +  81 

Or  X  =^  9  ±  1 

.'.     X  =  16,  or  2,  Ans. 

Page  294,  Art.  549. 

I.     a/i8«^  +  VJoa%^  —  yi^bV2ab  ±  scibV2ab,  Ans. 


collegeexami:n'atiox    problems.     267 

2,     Multiply  2  A/3  —  V  —  5 

By.  4V3  — 2V— 5 


54  —  4V  — 15 


4\/— 15  +  2  X  — 5 


(Arts.  312,  514.)        14  —  8\/  — 15,  Ans. 

^—  I       23  —  a;  4  +  ^ 

3.  Given =  7  — 

•^  7  5  4 

Clearing,  2o.^  —  20  +  644  —  28a:  =  980—140—352; 
Uniting  terms,  2'jx  =216 

.-.     X  =z  "8,  Ans. 

n-  ^'  —  3        ^  —  4  7 

4.  Given =  2V 

re  —  2       X  —  I 

Clearing  of  fractions, 
2o{x^—^x-\-^—x^  +  6x—8)  —  {x^  —  3.T  +  2)  7 

Or  40X —  100  =  ']X^  —  2i:6'  +  14 

Trans,  and  divid.,     x^  —  ^-f-x  —  —  ^^^ 

By  2d  method,  x 

Reducing,  ^  =  t4  ±  t^ 


T~" 


By  2d  method,  x  =  ii±V—^^  +  WT■ 


C.J  ,  -^    —    I4-i-I4 

.*.  o;  =  6,  or  2f,  ^4;?5. 

o       A   ^  2.<?  2  X  198 

5.     See  Art.  392.     n  =  — — -,  =  — -, =  9  ; 

^  ^^  rt  +  /        2  +  42 


Ins. 


O  A  7  ^    —     rt  42     2  . 

See  Art.  391.    a  = =  =  5,  \ 

■^  — I         9—1  / 

—  1  ^^~^  —  I     i~^  _       \. 

^^    —    ^3  >        ^^^  —    T  ^         -         —  9  5 


n  —  I     n — 2  ,     i — 2 


w  X X =  — i  X  - —  =:  — i  X  — I  =  ^ ; 

23-  3 

(a^—h^y  =  f^ ^^^ -—i-\'  etc.,  Jw5. 

^  ^  3ft^     9«='      8ifr 


2G8        COLLEGE     EXAMIN'ATION     PROBLEMS. 

7.  (a^  —  20^1  +  aWfi  =  a/(^^  —  2ab  +  1?)  a 

=  (a  —  b)  Vci,  Ans. 

8.  Let  Xy  y  and  z  be  the  times  required  by  A,  B,  and  0. 

.      Then  i  +  i  =  ^4^  =  ^        (:) 

Part  A  and  B  do  in  i  day ; 


X       y 


Part  B  and  C  do  in  i  day ; 


Part  A  and  C  do  in  i  day ; 

Adding  and  reducing,         -  +  -  +  -  =  i§  (4) 


Part  A,  B  and  C  do  in  i  day ; 

Subtracting  (2)  from  (4), 
Part  A  does  in  i  day; 

Subtracting  (3)  from  (4), 
Part  B  does  in  i  day ; 

Subtracting  (i)  from  (4), 
Part  C  does  in  i  day. 

Clearing  of  fractions,  a:  ==  10  days,  A's  time  ; 


I       I 

y 

7 

30 

I     I 

X        z 

I     . 

6 

/s 

I       I 

X      y 

+ 

I 

z 

10 

3^0 

I 
X 

A 

• 

I 

y 

i 

I 

z 

A 

(2) 


(3) 


;) 


y  zzi    6  days,  B's  time 
2;  =  15  days,  C's  time.    ,'  Ans. 
If  all  united  do  \  in  i  day,  it  will  take 
3  days  to  finish. 

3  =  First  number  ;      \ 

5  =  Second  number  ;  ,-  A71S. 

7  =r  Third  number,      ) 

See  solution  of  Ex.  6,  under  Art.  548. 


COLLEGE     EXAMINATION     PROBLEMS.       2G9 

Page  294,  Art.  550, 

I.    Given  -  +  -  =  2  (i) 

(3) 


I 

+ 

I 

— 

2 

X 

y 

I 

I 

H- 

- 

3 

X 

;2 

I 

I 

— 

+ 

— 

3 

y 

z 

Add  and  div.  by  2,      -i 1-  -  =  4  (4) 

''       X      y      z 

Subtracting  (3)  from  (4),      -  =  i,  or  a;  =  i  ; 

«  (2)         "  -  =  I,  or  y  =  i\)  Ans. 


«•  (i)         "  -  =  2,  or  2  =  I 


;2 


3  8  —  ^        ^—11 

2.     Given  7; =  

^  —  X  3  12 

Clearing  of  fractions, 

36  —  256  +  642-  —  4.1-2  —  i(^x  —  x^  —  88 

Uniting  and  divid.,   x^  —  15.?'  =  —  44 


By  2d  method,  x  =  -\^-  ±a/  — 44  +  H^ 

Reducing,  x  =  -^/  ±  | 

.♦.     X  =^  II,  or  4,  ^w.9. 

3.     Factoring  the  given  quantities,  we  have 
x'^  -^  4X  —  21  =  {x  +  7)  {x  —  3) 

X2  _      X-  S6    ^    {X  +   7)  (^^  -  8) 

Hence  f/-  c  ^Z.  =  a;  +  7  ;  ) 

And  I.  c.  m.  =  {x  +  l){x  —  3){x  —  ^)^  Ans. 

=  .,-3  _  4.^2  _  53:r  +  168,     ) 


270       COLLEGE     EXAMINATION^     PROBLEMS. 

4.     Let  .r  =  No.  of  days  B  requires  ; 

Then  a;  +  10  =  "         A       " 

By  condition, 

-  H =  T2J  part  A  and  B  do  in  i  day. 

X       X  -^  10  ^  "^ 

Clearing  of  frac,     12.T+  120  + i2.r  =  x-  4-  102: 

Or  x^  —  14X  =  120 


By  2d  method,  '  x  =^  j  ±  ^120  +  49 

Eeducing,  ^  =  7  ±  13 

.*.     a;  =  20  days,  B's  time ;  ) 
And  a:  +  10  =  30  days,  A's  time,  f 

See  Arts.  270,  271. 

Powers  of  y,  y^-\-     y^-\-     y'^-\-       y  -f    i 

Powers  of  3,  13  9  27     81 

Coefficients,  146  41 

^  =       (y  +  3)'  =  ^4+ 12^/3  +  54^2+ io8?/  +  8i 
—x^  =  —{y  +  3)^  =     —    y^—  9?/—  27?/— 27 

2X^  =       2(?/  +  3)2=z  2/+     12?/+l8 

—  3  = -  3 

;r4— a:3  4-2.r2_3  —  ^4_|_  ji^3_|_4y2/24-   g^y-^-GgyAns 


.^ 


m  35912 


M577055 


QA153 
T485 
Educ. 
Lib . 


